Er , Tm , and Lu ) : synthesis , molecular structure , and quantum motion of the endohedral nitrogen atom †

Systematic exploration of the synthesis of mixed-metal Dy–M nitride clusterfullerenes (NCFs, M = Gd, Er, Tm, Lu) is performed, and the impact of the second metal on the relative yield is evaluated. We demonstrate that the ionic radius of the metal appears to be the main factor allowing explanation of the relative yields in Dy–M mixed-metal systems with M = Sc, Lu, Er, and Gd. At the same time, Dy–Tm NCFs show anomalously low yields, which is not consistent with the relatively small ionic radius of Tm but can be explained by the high third ionization potential of Tm. Complete separation of Dy–Gd and Dy–Er, as well as partial separation of Dy–Lu M3N@C80 nitride clusterfullerenes, is accomplished by recycling HPLC. The molecular structures of DyGd2N@C80 and DyEr2N@C80 are analyzed by means of single-crystal X-ray diffraction. A remarkable ordering of mixed-metal nitride clusters is found despite similar size and electronic properties of the metals. Possible pyramidalization of the nitride clusters in these and other nitride clusterfullerenes is critically analyzed with the help of DFT calculations and reconstruction of the nitrogen inversion barrier in M3N@C80 molecules is performed. Although a double-well potential with a pyramidal cluster structure is found to be common for most of them, the small size of the inversion barrier often leads to an apparent planar structure of the cluster. This situation is found for those M3N@C80 molecules in which the energy of the lowest vibrational level exceeds that of the inversion barrier, including Dy3N@C80 and DyEr2N@C80. The genuine pyramidal structure can be observed by X-ray diffraction only when the lowest vibrational level is below the inversion barrier, such as those found in Gd3N@C80 and DyGd2N@C80. The quantum nature of molecular vibrations becomes especially apparent when the size of the inversion barrier is comparable to the energy of the lowest vibrational levels.


Introduction
Nitride clusterfullerenes (NCFs), i.e. metallofullerenes with endohedral M 3 N clusters, are the most abundant and versatile family of endohedral clusterfullerenes. 1 Since the discovery of the first NCF, Sc 3 N@C 80 , in 1999, 2 a large variety of NCFs with various fullerene cage sizes and different metal ions (M = Sc, Y, or lanthanides) have been synthesized. 3 In the M 3 N cluster, three M 3+ ions are located at the vertices of a triangle with the nitride ion N 3− in its center. The cluster transfers six electrons to the surrounding carbon cage. The fullerene C 80 -I h with a particularly stable hexaanion is the most suitable host for the endohedral species donating six electrons to the cage, 4 and for M = Sc, Y, and lanthanides from Gd to Lu the M 3 N@C 80 -I h NCFs are obtained with the highest yield among other cage sizes. However, when the cluster's size is incommensurate with the C 80 -I h cage, as is observed for the early lanthanides from La to Nd with large ionic radii, the distribution of the fullerene sizes shifts to larger cages and the total NCF yield is decreased substantially. 5 For Gd and Tb, although Gd 3 N@C 80 -I h and Tb 3 N@C 80 -I h are still the most abundantly produced Gd-and Tb-NCFs, the large size of the nitride cluster leads to substantial inner strain, which is partially compensated via a considerable pyramidalization of the M 3 N cluster inside the C 80 -I h cage. 3d,6 For smaller metals, the M 3 N cluster in M 3 N@C 80 -I h is planar or nearly planar. 3d,6a Thus, the balance between the ionic radius of the metal and the fullerene size is an important factor influencing the yield and molecular structure of NCFs. 1b,c,6b Combining metals of different sizes within one NCF molecule, which would be then dubbed "mixed-metal NCF", is an efficient approach to vary the cluster size and hence alter the product distribution of NCFs. As the ionic radius of Sc is considerably smaller than those of lanthanides, Sc has often been used in the synthesis of binary mixed-metal NCFs. Sc-Y, 7 Sc-Ti, 8 and Sc-V, 9 all Sc-lanthanide binary systems forming NCFs, and even two ternary-metal NCFs, ScYErN@C 80 and DyErScN@C 80 , have been produced. 10 A number of non-Sc mixed-metal NCFs have also been synthesized, including Ti-Y, 11 Ce-Y, 10e,f Ce-Lu, 12 Gd-Ho, 13 Gd-Lu, 13,14 Ho-Y, 15 Ho-Lu, 14,15 and Lu-Y. 16 In addition to the alteration of yields and product distribution in comparison with corresponding homometallic NCFs, mixedmetal NCFs may also exhibit different chemical, 7,17 electrochemical, 10e,f,12a and magnetic properties. 15,18 The structural diversity of nitride clusters offers a possibility to tune the magnetism of lanthanide-NCFs. The nitride ion in the center of the M 3 N cluster bears a large negative charge (the formal charge is −3e; the QTAIM analysis gives the value of ca −1.7e). 19 Besides, NCF molecules feature rather short lanthanide-nitrogen bonds. The combination of these two factors leads to a large negative charge from a nitride ion in close proximity to a lanthanide and creates a strong axial ligand field (LF) in NCFs, which results in an easy-axis magnetic anisotropy for Ce, Pr, Nd, Tb, Dy, and Ho, and an easyplane anisotropy for Er and Tm ions. 10i Ab initio calculations performed for Dy-NCFs predict very large LF splitting of Dy 3+ total momentum ( J) states in the range of 1500 cm −1 , 20 and all three Dy x Sc 3−x N@C 80 -I h NCFs (x = 1-3) were found to be single molecule magnets. 18a,21 However, due to the different cluster compositions and intramolecular Dy⋯Dy interactions, DySc 2 N@C 80 -I h , Dy 2 ScN@C 80 -I h , and Dy 3 N@C 80 -I h exhibit substantially different magnetic properties, which shows that combining Dy with other metals in nitride clusterfullerenes may lead to even more interesting magnetic phenomena, and further exploration of Dy-based mixed-metal NCFs is therefore an important task.
In this work we focus on the binary Dy-metal NCFs with a C 80 -I h cage (Fig. 1) and study the influence of the non-Dy metals in the cluster on the synthesis and molecular structure of mixed-metal NCFs, keeping in mind further studies of their magnetic properties, which will be reported elsewhere. As a partner for Dy we have chosen the following elements: Lu as the lanthanide with the smallest ionic radius (Shannon's ionic radius 22 R 3+ = 0.86 Å); Gd as the largest lanthanide (R 3+ = 0.94 Å), still favoring the formation of M 3 N@C 80 -I h ; and Er (R 3+ = 0.89 Å) and Tm (R 3+ = 0.88 Å) as lanthanides with intermediate sizes. Besides, these lanthanides have different magnetic states in NCFs, 10i and hence their combination with Dy may lead to diverse magnetic properties of mixed-metal NCFs. In the following, we first present a systematic analysis of the influence of the metal on the relative yields of mixed-metal Dy NCFs, and then an analysis of how a combination of Dy with lanthanide ions of different sizes affects the molecular structure and in particular the shape of the trimetal-nitride cluster. In the end we demonstrate how quantum vibrational effects dramatically affect the outcome of the diffraction experiments.

Synthesis
The synthesis of all mixed-metal EMFs in this work was performed following the same protocol. Graphite rods were coredrilled and filled with a mixture of Dy/M (1 : 1 molar ratio), graphite, and melamine as a source of nitrogen. The molar ratio Dy : M : N : C was kept at 1 : 1 : 10 : 15. The rods were evaporated under a He atmosphere (200 mbar) using a current of 100 A. Current phases were 30-60 s long, followed by a cooling period. As only the cathode is evaporated in the arc-discharge, the polarity of the rods was changed between the phases to ensure evaporation of both electrodes. The soot produced was collected, washed with acetone and extracted with CS 2 for 20 hours. Then the CS 2 was evaporated, the fullerene extract was redissolved in toluene, characterized by analytical HPLC and LDI mass-spectrometry, and then separated by multi-step HPLC as described below.
Comparison of the binary Dy-M NCF systems Fig. 2 compares chromatograms of the raw fullerene extracts obtained for Dy-Gd, Dy-Er, Dy-Tm, and Dy-Lu systems. For all systems, chromatograms are dominated by a strong peak with a retention time near 32 min, which based on our previous experience with NCF synthesis is assigned to the mixture of Dy x M 3−x N@C 80 -I h NCFs (x = 0-3). The peak next to it with a retention time of ca. 34 min is assigned to NCFs with a C 80 -D 5h cage, whereas the peaks at longer retention times are NCFs with larger cages. Note that the relative yield of NCFs with larger cages is the highest in the Dy-Gd system. Nonetheless, as discussed in the introduction, NCFs with C 80 -I h cages are produced with the highest yield, and in the following we will focus only on this cage isomer. Unless otherwise stated, from here on we will omit the cage symmetry designation and assume C 80 to be C 80 -I h .
For an equimolar mixture of Dy and a metal M in the starting material (i.e. the Dy : M molar ratio is 1 : 1), the ratio of Dy 3 N : Dy 2 MN : DyM 2 N : M 3 N clusters in the synthesized NCF mixture should be 1 : 3 : 3 : 1 given that the product ratio follows binominal statistical distribution without a bias for any of the metals. However, the studies of mixed-metal EMFs showed that the ratio of the cluster composition rarely follows the expected statistical distribution. 2,10e,h,23 The use of an array of metals in combination with Dy in this work enables a systematic study of this phenomenon. As a method for fast analysis of the sample composition, here we use laser-desorption ionization mass-spectrometry (LDI-MS). Although LDI-MS can hardly be applied for a quantitative estimation of an arbitrary fullerene mixture because of different ionization efficiencies, it is known that all M 3 N@C 80 NCFs have almost identical oxidation potentials independent of the metals in the cluster, 1d and hence similar ionization efficiency in the positive ion mode can be expected for all M 3 N@C 80 molecules studied in this work. Thus, the analysis of the LDI-MS intensities should give if not the quantitative estimation of the relative yield in each mixed-metal system, but at least a trend in the values. Fig. 2b and 3 show mass-spectra of the M 3 N@C 80 fraction for each Dy-M system. As Dy and Lu have sufficiently different atomic masses, the peaks of all four Dy x Lu 3−x N@C 80 (x = 0-3) species can be easily distinguished in the mass spectrum (Fig. 2b). For other Dy-M systems, the relatively broad isotopic distribution and closer atomic masses result in overlapping spectral patterns, which still can be used for the estimation of the sample composition by matching with theoretical isotopic distributions for the involved species ( Fig. 3 and S1 †). The relative yield of a given Dy x M 3−x N@C 80 compound in the Dy-M mixed system referred to the yield of Dy 3 N@C 80 is then computed as a ratio of the total LDI-MS intensities of Dy x M 3−x N@C 80 to Dy 3 N@C 80 (Table S1 †). The relative yield of Dy 3 N@C 80 in each system is then considered to be equal to 1. The relative yields estimated from mass-spectral data are plotted in Fig. 4 along with the data on the Dy-Sc system studied in our group earlier. A strong deviation from the binominal statistical distribution (1 : 3 : 3 : 1) is found in all binary systems. Note that the following discussion is based on the relative yields of NCFs in mixed-metal systems. The absolute yield of Dy 3 N@C 80 varies in different mixed-metal systems, and hence Fig. 4 and the data in Table S1 † do not give an estimation of the total yields of NCFs. We do observe a general trend that the absolute yield of Dy 3 N@C 80 in the mixed-metal systems with a small second metal (Sc or Lu) is considerably lower than the absolute yield of Dy 3 N@C 80 in a mixed-metal system with bigger metals (such as Gd), but a precise estimation of the absolute yields is not possible due to the small amount of EMFs produced by arc-discharge synthesis and is not discussed further.
A clear tendency of the relative yield variation with the size of the metal is seen in Fig. 4. Sc with the smallest ionic radius gives the highest relative yields of Dy 2 MN@C 80 , DyM 2 N@C 80 , and M 3 N@C 80 . The ratio of Dy 2 ScN and DySc 2 N is 0.5 instead of the expected 1, and the ratio of DySc 2 N and Sc 3 N is 1.5 instead of 3. Clearly, under identical conditions, Sc is much more predisposed to the formation of M 3 N@C 80 than Dy. In the Dy-Lu and Dy-Er systems the relative yields of DyM 2 N@C 80 and M 3 N@C 80 are also much higher than the expected 3 to 1, respectively, but smaller than those for Sc analogs. The decrease of the relative yield of mixed-metal species is in line with the increase of the ionic radii from Sc (0.75 Å) to Lu (0.86 Å) to Er (0.89 Å). For Gd (0.94 Å), whose ionic radius is higher than that of Dy (0.91 Å), the situation is reversed, and the relative yields of Dy 2 GdN, DyGd 2 N, and Gd 3 N species decrease with the number of Gd ions in the cluster. Thus, a simple conclusion might be drawn based on these data: the metals with an ionic radius smaller than Dy are more favorable for NCF formation, whereas the metals with a bigger ionic radius are less predisposed to NCF formation. However, the data for the Dy-Tm system shows that the ionic radius is not the only factor affecting the yield of NCFs.
With an ionic radius of 0.88 Å, Tm 3+ is slightly smaller than Er 3+ and considerably smaller than Dy 3+ , but the distribution of Dy-Tm NCFs points to the higher preference for Dy versus Tm for NCF formation. The relative yields of Dy 2 TmN@C 80 and DyTm 2 N@C 80 are even smaller than those of Dy 2 GdN@C 80 and DyGd 2 N@C 80 . We suggest that the reason for the anomalously low yield of Dy-Tm NCFs lies in the electronic properties of Tm, particularly in its third ionization potential. It is known that in monometallofullerenes some lanthanides are divalent, whereas others adopt a trivalent state. Shinohara et al. demonstrated that the valence states of lanthanides in EMFs correlate well with their third ionization potentials (IP 3 , see also Fig. S2 †). 24 The metals with a large IP 3 exceeding 23 eV, namely Sm, Eu, Tm, and Yb, donate only two electrons to the fullerene cage and remain divalent, whereas IP 3 values of La, Ce, Pr, Nd, Gd, Tb, Dy, Ho, Er, and Lu are below the threshold of 23 eV. These lanthanides are ionized by a hosting fullerene cage to their trivalent state. Furthermore, the lanthanides with an IP 3 below 23 eV form nitride clusterfullerenes (in which the valence state of the metal is also trivalent), whereas clusterfullerenes of Sm, Eu, and Yb are not known. Tm is the only lanthanide able to adopt a variable valence state in EMFs, a divalent state in monometallofullerenes and a trivalent state in NCFs. 25 Presumably, the high IP 3 value of Tm still takes its toll in NCF formation by lowering the synthetic yields of Dy 2 TmN@C 80 , DyTm 2 N@C 80 , and Tm 3 N@C 80 .
As the yield of Dy-Tm NCFs is too low, further accumulation and separation of Dy x M 3−x N@C 80 NCFs were only performed for M = Gd, Er, and Lu. Dy x M 3−x N@C 80 NCFs with the same fullerene cage show very similar retention behavior, and their separation requires several steps of recycling HPLC. Fig. 5 shows plots of recycling chromatograms for all three mixtures  Gd) in each mixed-metal system determined from LDI-MS intensities and normalized to the intensity of Dy 3 N@C 80 -I h (note that the data are plotted in a logarithmic scale). Also shown are the yields expected for the binominal statistical distribution of metals in the nitride cluster (1 : 3 : 3 : 1). Numerical data used to prepare this plot can be found in Table S1 in the ESI. † Note that the absolute yield of Dy 3 N@C 80 varies in different mixed-metal systems.
obtained for the main Dy x M 3−x N@C 80 fractions highlighted in Fig. 2 at the first step ( Fig. S3-S8 † show further separation details). The Dy-Gd system offers the most straightforward separation among the three. Clear resolution into four components (with Dy 3 N, Dy 2 GdN, DyGd 2 N, and Gd 3 N clusters) can be seen after 16 cycles. For the Dy x Er 3−x N@C 80 mixture, a similar level of separation requires at least twice more cycles ( Fig. 5 and S5 †), which further complicates the separation not only because of the increased time demands, but also because of the overlap of the fastest fractions at the nth cycle with the slowest fractions at the (n−1)th cycle. Surprisingly, the Dy x Lu 3−x N@C 80 mixture proved to be the most difficult to separate despite the largest difference in the ionic radii of Dy 3+ and Lu 3+ . Whereas the resolution of the Dy 3 N@C 80 and Dy 2 LuN@C 80 HPLC peaks is possible after 25 and 40 cycles, respectively ( Fig. 5 and S7 †), the retention times of Lu 3 N@C 80 and DyLu 2 N@C 80 are so close that their separation cannot be accomplished even after 70 cycles (Fig. S7 and 8 †).
In summary, multistep recycling HPLC allowed the isolation of Dy 2 GdN@C 80 , DyGd 2 N@C 80 , DyEr 2 N@C 80 , and Dy 2 ErN@C 80 in high compositional purity exceeding 90-95% based on mass-spectroscopic analysis (see Fig. S4, S6, and S8 †). The retention behavior of Dy 2 LuN@C 80 , DyLu 2 N@C 80 and Lu 3 N@C 80 was found to be too similar to offer efficient separation without a substantial loss of the material. Dy 2 LuN@C 80 could be separated to ca. 90% purity (the sample also contained ca. 10% DyLu 2 N@C 80 and ca. 3% Lu 3 N@C 80 ), whereas the separation of DyLu 2 N@C 80 from Lu 3 N@C 80 is hardly possible and was aborted when the relative content of the two NCFs in the sample was ca. 1 : 1.35. Note that earlier we have had similar difficulties with the separation of the Ho x Lu 3−x N@C 80 NCFs. 15

Spectroscopic characterization
Since the absorption of UV and visible light by EMFs are dominated by the π → π* transition of the fullerene cage, UV-Vis absorption spectra of M 3 N@C 80 NCFs are known to be very similar irrespective of the endohedral M 3 N cluster, unless the latter also includes Sc atoms, in which case the differences between spectra are more pronounced. 1d,10g,k,16,23 Indeed, UV-Vis absorption spectra of the Dy-Gd and Dy-Er M 3 N@C 80 families also follow this trend and have an almost identical appearance with absorptions near 325, 410, and 555 nm and 3-4 weak features in the long-wavelength range at 620-720 nm (Fig. 6). However, more detailed analysis of the lowest-energy part of the spectra shows a small but systematic variation of the peak energies. Upon going from Gd 3 N@C 80 to Dy 3 N@C 80 and then to Er 3 N@C 80 via corresponding mixed-metal counterparts (i.e. with a gradual decrease of the cluster size), absorption energies shift to shorter wavelengths (higher energies) from 708/ 675 nm in Gd 3 N to 695/665 nm in Er 3 N@C 80 (Fig. 6, inset). Although the HOMO and LUMO of non-Sc M 3 N@C 80 -I h NCFs are mainly localized on the fullerene cage, a small contribution of the metal cluster to the LUMO was also established, 26 which is presumably responsible for the blue-shift of the lowest energy transitions with the decrease of the cluster size.

Single-crystal analysis and molecular structures
The molecular structures of DyEr 2 N@C 80 and DyGd 2 N@C 80 were established by single-crystal X-ray diffraction using syn- Fig. 6 Overlaid UV-Vis absorption spectra of isolated M 3 N@C 80 NCFs measured in toluene solution at room temperature. The inset shows an expansion of the low-energy part; the spectra are offset for easier comparison. chrotron radiation at BESSY II of the Helmholtz-Zentrum Berlin. 27 Single crystals suitable for diffraction analysis were obtained by co-crystallization of DyEr 2 N@C 80 and DyGd 2 N@C 80 with Ni II (OEP) (OEP = octaethylporphyrin). The DyM 2 N@C 80 ·Ni II (OEP) fragments of DyGd 2 N@C 80 ·Ni II (OEP)·2 (C 6 H 6 ) and DyEr 2 N@C 80 ·Ni II (OEP)·2(C 6 H 6 ) crystals are shown in Fig. 7. As in many other M 3 N@C 80 ·Ni II (OEP) co-crystals, both fullerene cages and Ni II (OEP) molecules are well ordered with their nearest cage carbon-Ni contacts of 2.81-2.82 Å.
In the icosahedral C 80 -I h cage the M 3 N cluster has many energetically equivalent positions, which result in the free rotation of the cluster inside the cage in solution and may lead to disorder in the crystalline phase; even stronger disorder may be expected for the mixed-metal NCFs due to the overlap of different metal positions in the cluster. The unique advantage of co-crystallization with Ni II (OEP) for NCFs is that not only the fullerene cage, but also the endohedral cluster is often well ordered in the crystal. 6b,10c,j The plane of the M 3 N cluster is usually perpendicular to the porphyrin plane of the Ni II (OEP) molecule, two metals face the Ni II (OEP)-coordinated part of the cage, and the third metal is oriented towards the opposite side of the fullerene cage (Fig. 7). In the MSc 2 N@C 80 ·Ni II (OEP) and M 2 ScN@C 80 ·Ni II (OEP) co-crystals, the position farthest from the porphyrin is always adopted by the larger lanthanide, whereas one or two Sc atoms are oriented closer to the Ni II (OEP). Likewise, in the CeLu 2 N@C 80 ·Ni II (OEP)·2C 7 H 8 crystal (CeLu 2 N@C 80 -I h is the only non-Sc mixed-metal NCF studied by single-crystal X-ray diffraction before this work), the larger Ce atom is located further away from the Ni II (OEP), and smaller Lu atoms are located closer to the porphyrin. 12b We have recently studied the ordering of the endohedral cluster in Sc 3 N@C 80 and YSc 2 N@C 80 by Ni II (OEP) theoretically and found that the ordering can be explained by simple electrostatic consideration based on the distribution of the electrostatic potential in the molecules. 28 The importance of the electrostatic potential distribution for co-crystallization with Ni II (OEP) has also been reported for some other empty and endohedral fullerenes. 29 Furthermore, the study of YSc 2 N@C 80 showed that different metals (Y and Sc in that case) cause slightly different distribution of electrostatic potentials outside the fullerene cage, which may lead to a further alignment of the mixed-metal clusters.
Although in the DyM 2 N@C 80 NCFs studied in this work the difference of the ionic sizes of metals is not as pronounced as in the already reported mixed-metal NCFs, co-crystallization with Ni II (OEP) still results in remarkable ordering of the nitride clusters. In DyGd 2 N@C 80 ·Ni II (OEP), the single Dy ion is fully ordered and is located remotely from the Ni II (OEP), whereas Gd atoms are refined to two sites with occupancies of 0.86 and 0.14 ( Fig. 7). Two sites with occupancies of 0.58 and 0.42 located at a distance of 0.91(2) Å from each other are refined for nitrogen. For the main site of the DyGd 2 N cluster shown in Fig. 7c, the bond lengths are 2.034(10), 2.085(10), and 2.097(11) Å for Dy1-N1A, Gd1-N1A, and Gd2-N1A, respectively, and the bond angles at nitrogen N1A are: ∠Dy1N1AGd1 = 116.8(5)°, ∠Dy1N1AGd2 = 118.0(5)°, and ∠Gd1N1AGd2 = 111.1(5)°with a sum of 345.9°. The geometry of the minor site of the DyGd 2 N cluster shown in Fig. S9b † has a small deviation from the major site geometry. Splitting of the nitrogen positions and deviation of the sum of ∠MNM' angles from 360°indicate that the DyGd 2 N cluster inside the C 80 -I h cage is pyramidal.
In DyEr 2 N@C 80 ·Ni II (OEP), two sites of DyEr 2 are well recognized with occupancies of 0.8 and 0.2. The Er1 atom is located far from the Ni II (OEP). The Er2 and Dy1 atoms are overlapped and face the porphyrin-coordinated site of the cage. Thus, in both DyGd 2 N@C 80 ·Ni II (OEP) and DyEr 2 N@C 80 ·Ni II (OEP) crystals, the position farthest from the porphyrin is occupied by the smaller lanthanide (Dy or Er), in contrast to the situation described earlier for MSc 2 N,10c,d,j,21b,30 M 2 ScN,10j,21c and CeLu 2 N 12b clusters.
Note that Dy/Gd or Dy/Er are crystallographically rather similar and it is hardly possible to distinguish them unambiguously. For DyGd 2 N@C 80 , our assignment shown in Fig. 7b is based on the higher electron density at the Dy1 site when compared to both Gd sites. Besides, the shorter Dy-N and longer Gd-N bond distances determined with this assignment (Fig. 7c) are in good agreement with the larger ionic radius of Gd 3+ in comparison with that of Dy 3+ . Likewise, assignment of the Er1(A/B) site in DyEr 2 N@C 80 is also based on the higher electron density at that position and on the reasonable distribution of the metal-nitrogen bond lengths following the expected shorter Er-N and longer Dy-N bonds (Fig. 7f ). However, current experimental data do not allow us to exclude some overlap (in the range of 10%) between Gd and Dy positions in DyGd 2 N@C 80 or a further overlap between Er1 and Dy in DyEr 2 N@C 80 .
The pronounced pyramidalization of the DyGd 2 N cluster and the nearly planar shape of the DyEr 2 N cluster are in line with the crystallographic studies of other M 3 N@C 80 molecules. 6b The shape of the nitride cluster in NCFs depends on the match between the size of the cluster and the cage size. The most stable form of the cluster is planar, and it is realized in those NCFs which offer sufficient inner space for the cluster. If the cage size is fixed as in the M 3 N@C 80 series considered here, the increase of the metal size leads to the inner strain of the structure. The geometrical parameters of the M 3 N cluster have a certain flexibility, and the M-C and M-N bonds may become shorter in comparison with their optimal values, to maintain the planarity of the cluster. However, when the planarity of the cluster requires shortening of the M-N and M-C bonds beyond a certain threshold, the nitrogen atom is pushed out of the M 3 plane and the cluster adopts a pyramidal shape instead. Table 1 summarizes the Gd-N, Dy-N, Er-N, and Lu-N bond lengths in different M 3 N@C 80 NCFs determined by singlecrystal X-ray diffraction studies. The average M-N bond length in Lu 3 N@C 80 , Er 3 N@C 80 , and Dy 3 N@C 80 is in the range of 2.04-2.05 Å, although the ionic radii of Dy 3+ and Lu 3+ are different by 0.05 Å. The Er 3 N and Lu 3 N clusters are nearly planar, whereas for Dy 3 N the experimental data allow two alternative interpretations: one suggests a planar cluster and a single nitrogen site, in which the thermal ellipsoid is strongly elongated in the out-of-plane direction. 31 A more recent interpretation of the same diffraction data suggests splitting of the nitrogen position between two sites leading to a pyramidal Dy 3 N cluster. 6b In Gd 3 N@C 80 , a further increase of the metal size results in an average Gd-N bond length of 2.08 Å, and the Gd 3 N cluster has to adopt a pyramidal configuration in M 3 N@C 80 . 6a Note that Dy-N bonds of 2.0-2.1 Å determined for Dy-containing NCFs are much shorter than typical Dy-N bond lengths. Fig. 8 shows the distribution of 5369 Dy-N bond lengths in molecular Dy compounds retrieved from the Cambridge Structural Database (CSD). The shortest non-fullerene Dy-N bond reported to date is 2.14 Å long, 32 whereas in a majority of such compounds the Dy-N bond lengths are in the range of 2.3-2.7 Å. 33 The short Dy-N bond length is an indication of significant strain experienced by lanthanide M 3 N clusters encapsulated inside the C 80 fullerene cage.
Mixed-metal nitride clusters allow redistribution of the bond lengths and hence a partial release of the strain. Geometrical parameters of MSc 2 N and M 2 ScN clusters (M = Gd, Dy, Er) clearly illustrate this effect. M-N bonds in MSc 2 N@C 80 NCFs are considerably longer than the M-N bonds in the corresponding M 3 N@C 80 molecules (Table 1). In M 2 ScN@C 80 (M = Gd, Dy) the M-N bond lengths become shorter than those in MSc 2 N clusters and approach the values found in M 3 N@C 80 molecules. Yet, both MSc 2 N and M 2 ScN clusters are planar for all lanthanides studied so far. 10c,d,j,21b,c,30 In DyGd 2 N@C 80 , the length of the Gd-N bonds is comparable to those in Gd 3 N, and the Dy-N bond is even somewhat shorter than that in Dy 3 N@C 80 . Thus, both Gd-N and Dy-N bonds reached their shortest threshold values, and the DyGd 2 N cluster has to become pyramidal to release the strain further. For DyEr 2 N the situation is less obvious because of the overlapped Dy/Er positions, which make the bond length estimation unreliable. For the well-modeled Er1 atom in DyEr 2 N@C 80 , the Er1A-N bond is noticeably shorter than that in Er 3 N@C 80 , which indicates that the Dy-N bond can release its strain by elongation at the expense of the shortening of the Er-N bonds.
As a measure of the pyramidalization of the M 3 N cluster it is convenient to use the displacement of the nitrogen atom from the metal plane (Table 2). For instance, in Gd 3 N@C 80 and Tb 3 N@C 80 nitrogen atoms are displaced above/below the M 3 plane by 0.522/0.463 and 0.453/0.405 Å, respectively (the nitrogen atom in these molecules is split between two sites, hence two different values). 3d,6a In DyGd 2 N@C 80 this displacement distance is 0.456/0.452 Å, somewhat smaller than that in Gd 3 N but larger than that in the Tb 3 N cluster. If the split-nitrogen model of the Dy 3 N@C 80 data is adopted, the out-of-plane displacement of nitrogen in Dy 3 N would be 0.26 Å (here the value is calculated as a half of the distance between two nitrogen sites). I.e., pyramidalization is decreased considerably from Gd 3 N over Tb 3 N to Dy 3 N.
For a deeper understanding of these phenomena, we also studied the pyramidalization of the nitride cluster in a series of lanthanide and mixed-lanthanide M 3 N@C 80 molecules theoretically. DFT calculations were performed at the PBE-D level with a plane-wave basis set and corresponding projector augmented-wave potentials, treating 4f-electrons as a part of the core as implemented in the VASP 5.0 package. 36 The computed out-of-plane displacements of nitrogen for Gd 3 N@C 80 , DyGd 2 N@C 80 , and Tb 3 N@C 80 are 0.468, 0.396, and 0.365 Å   respectively. Notably, for the midsize cluster in Dy 3 N@C 80 calculations predict the value of 0.203 Å, but already in DyEr 2 N@C 80 the cluster is nearly planar, with the nitrogen displacement amounting to only 0.057 Å (compare to the experimental result of 0.05(1) Å). Similarly, for Dy 2 GdN@C 80 and Dy 2 ErN@C 80 the calculation predicts a pyramidal cluster with a displacement parameter of 0.311 and 0.086 Å, respectively. Finally, nearly planar cluster geometries are also found for Er 3 N@C 80 (0.020 Å) and Lu 3 N@C 80 (0.007 Å). These computational results are in agreement with the trends predicted in other theoretical studies of M 3 N@C 80 molecules. 13,37 Overall, DFT calculations support well the experimental observations of the strong cluster pyramidalization for Gd 3 N and DyGd 2 N, a moderately pyramidal cluster geometry in Dy 3 N@C 80 , and a nearly planar cluster in DyEr 2 N@C 80 . However, as we show below, a direct comparison of X-ray data and results of DFT calculations may be misleading if the real physical quantities behind diffraction data are not considered correctly.

Cluster pyramidalization and vibrational displacements
Structural analysis of NCFs such as that given in the previous section or in numerous publications on this subject (see ref. 6b for a systematic analysis of M 3 N@C 80 structures) implicitly assumes that X-ray diffraction provides the ground state geo-metry of the molecule. Strictly speaking, this is not entirely correct. Thermal motion of atoms and population of excited vibrational levels may significantly distort the observed structure, especially if the potential energy surface is rather shallow. Since these effects may affect the interpretation of the diffraction data substantially, we decided to analyze in depth how vibrational degrees of freedom may influence the structure of the nitride cluster.

Harmonic approximation
DFT-based vibrational analysis of M 3 N@C 80 NCF molecules (M = Y, Gd-Er, Table 2) shows that vibrations of the nitrogen atom in these molecules are well separated from vibrational displacements of the other atoms. Three degrees of freedom of the nitrogen atom produce two vibrational modes of the M 3 N cluster. The displacement of the nitrogen perpendicular to the plane of metals forms the cluster deformation mode (γ M 3 N ), whereas the in-plane motion constitutes the two-fold degenerate antisymmetric M-N stretching mode (ν asM-N ). The ν asM-N mode occurs at frequencies of 700-750 cm −1 and is easily detectable in IR spectra due to its medium-strong intensity. 31,39 The out-of-plane oscillation frequency predicted in the harmonic approximation depends strongly on the metal size and ranges from 256 cm −1 in Sc 3 N@C 80 (exp. value is 236 cm −1 , ref. 40) over 145 cm −1 in Lu 3 N@C 80 and down to 51 cm −1 in Ho 3 N@C 80 , for which DFT predicts the lowest γ M 3 N frequency in the whole M 3 N@C 80 series ( Table 2). The fre-  (8) The molecules are listed in the descending order of their DFT-predicted pyramidalization (displacement of nitrogen from the metal plane). b l z is computed as a square root of l 2 in eqn (1), in which ν and μ are substituted with the DFT vibrational frequency of the γ M 3 N mode and atomic mass of nitrogen, respectively. c Single nitrogen site. d Nitrogen is split between two sites and pyramidalization is computed as a half of the distance between two nitrogen sites. e In the pyrrolidine monoadduct. quency variation from Sc to Lu and Ho reflects the trend of the M 3 N cluster to flatten its potential energy surface along the pyramidalization coordinate with the increase of the metal size. When the metal size increases beyond Ho, the cluster adopts a pyramidal shape with a double-well energy potential, and the calculated γ M 3 N frequency increases again from 72 cm −1 in Dy 3 N@C 80 to 128 cm −1 in Tb 3 N@C 80 and 143 cm −1 in Gd 3 N@C 80 .
The large difference of the γ M 3 N and ν asM-N vibrational frequencies results in a more pronounced out-of-plane displacement of the nitrogen atom in M 3 N@C 80 , leading to an elongation of the nitrogen thermal ellipsoid in the out-of-plane direction. In the harmonic approximation, which implies that the energy is a quadratic function of atomic displacements, the thermal displacements can be calculated as the root-meansquare vibrational amplitudes of a quantum harmonic oscillator: 41 where μ is a reduced mass, and ν is a vibrational frequency. Since in-plane and out-of-plane oscillations of nitrogen in M 3 N@C 80 are well separated, we can compute the out-of-plane (l z ) and in-plane (l x,y ) nitrogen amplitudes as a square root of 〈l 2 〉 in eqn (1), in which μ is substituted by the atomic mass of nitrogen, and ν is either the vibrational frequency of the γ M 3 N mode (giving l z ) or the frequency of the ν asM-N mode (giving l x,y ). At a temperature of 100 K often used in XRD studies of fullerene crystals, the calculated out-of-plane and in-plane nitrogen amplitudes are l z /l x,y = 0.074/0.045 Å in Sc 3 N@C 80 , 0.103/0.041 Å in Lu 3 N@C 80 , 0.259/0.040 Å in Ho 3 N@C 80 , and 0.187/0.040 Å in Dy 3 N@C 80 . Even at 0 K, zero-point motion for the oscillator with a frequency of 80 cm −1 has an amplitude of 0.123 Å, and with the increase of temperature to 100 K the amplitude may reach 0.170 Å due to the population of vibrational excited states (for ν = 80 cm −1 , the populations of the first three levels at 100 K are 0.68, 0.22, and 0.07). One should also note that in the excited states the probability density moves closer to the edges of the potential well, as opposed to the center of the well for the zeroth level, which increases the probability to find the nitrogen atom above and below the metal plane even if the cluster has a planar geometry in the ground state. To summarize, rather large vibrational displacements of the nitrogen atom may potentially lead to erroneous conclusions on the perceived cluster geometry. When the cluster is on the edge between planar and pyramidal shapes, the potential energy surface is flat, and the vibrational frequency of the outof-plane deformation is low. This defines the large vibrational amplitudes of ca. 0.2 Å and partial localization of the probability density above and below the metal plane. The thermal ellipsoid of nitrogen should then be strongly elongated, and the question whether nitrogen should be treated as a single site ("planar cluster") or split into two sites ("pyramidal cluster") may become very ambiguous.

Beyond harmonic approximation
Simple harmonic approximation already points to the potentially complicated contribution of vibrational motion to the experimental molecular structure. But flat energy potentials and large vibrational amplitudes also indicate that the harmonic approximation may not be adequate by itself. A more reliable description of the system requires explicit consideration of the energy potential along the pyramidalization coordinate. With the use of DFT, we obtained potentials for five M 3 N@C 80 molecules by optimizing their structures with the nitrogen atom fixed at different distances above and below the plane of metal atoms (Fig. 9).
For the strongly pyramidal Gd 3 N cluster in Gd 3 N@C 80 , the energy profile (E(z)) has a classical double-well potential shape with the minima for the nitrogen atom at 0.45 Å above and below the Gd 3 plane (Fig. 10). Quite common in many fields of Fig. 9 (a) Molecular structure of M 3 N@C 80 with a pyramidal cluster and definition of the axis z used in the analysis; an equivalent position of the nitrogen atom below the plane of metal atoms is semitransparent. (b) Probability density for finding the nitrogen atom at a certain position along the z axis in different M 3 N@C 80 molecules computed for T = 100 K as the temperature often used in X-ray diffraction studies of EMFs.
analytical physics, the double-well potentials are well developed and understood since the studies of pyramidal inversion in NH 3 and related molecules. 42 The E(z) maximum at z = 0 Å, i.e. the pyramidal inversion barrier, is only 177 cm −1 in Gd 3 N@C 80 . The symmetrized ab initio potential profile (dots in Fig. 10) can be described well using an analytical function E(z) = Az 4 − Bz 2 (solid lines in Fig. 10). After fitting A and B parameters the Schrödinger equation is solved numerically using this analytical potential and a particle mass equal to that of a nitrogen atom (14 Da), which gives a quasi-degenerate ground state for Gd 3 N@C 80 with the symmetric and antisymmetric state energies of 71.2 and 73.2 cm −1 , respectively, and an inversion splitting of 2 cm −1 . The probability to find the nitrogen atom at a certain z-position (defined as a square of the wavefunction |Ψ n 2 | for the nth state) has two maxima closely coinciding with the potential minima. Thus, the pyramidal shape of the Gd 3 N cluster is well defined since the zeroth (|0〉) and first (|1〉) vibrational levels lie within the potential wells. However, already the second energy level at 174.5 cm −1 almost coincides with the inversion barrier, and |Ψ 2 2 | has a broad maximum at z = 0. At higher energy, the structure of the vibrational levels resembles that of a system with a harmonic potential, although the levels are not equidistant.
The experimentally observable positions of the nitrogen (e.g., from diffraction data) correspond to the maxima of the total probability density, which is temperature dependent because of the variation of thermal populations of the energy levels. The probability density at a temperature T can be obtained as a sum we find a well-behaving double maximum picture ( Fig. 9b and 10) with a slight temperature variation of the maximum position and a considerable broadening of the distribution. Yet two maxima are clearly seen up to room temperature ensuring that X-ray diffraction data give a correct description of the pyramidal Gd 3 N cluster. A double-well energy profile with an inversion barrier of only 69 cm −1 is found for DyGd 2 N@C 80 . The decrease of the barrier caused by the reduced size of the M 3 N cluster leads to a considerable change in the structure of the vibrational levels in comparison with those in Gd 3 N@C 80 . The ground vibrational level of DyGd 2 N@C 80 is found at 45.3 cm −1 , which lies within the double-well. However, the |Ψ 0 2 | function has a complex shape with two broad and strongly overlapping peaks, whose maxima are shifted from the positions of the potential minima to much smaller z values. Thus, if X-ray diffraction would be performed at helium temperature, an apparent pyramidalization deduced from diffraction data would be underestimated or even missed because the probability minimum at z = 0 is rather shallow. The first excited vibrational level of DyGd 2 N@C 80 is located at 63.7 cm −1 . Note that the quasidegeneracy of the |0〉 and |1〉 levels observed in Gd 3 N@C 80 is lifted here. The |Ψ 1 2 | distribution has a more common shape with two well-defined maxima near the positions corresponding to the minima of the energy profile. Considerable alteration of the shape of the |Ψ 0 2 | and |Ψ 1 2 | functions and a comparably small energy difference between the levels lead to pronounced changes in the probability density even after a modest temperature increase from 5 to 40 K: the maxima become sharper and shift further away from each other, whereas the minimum between them becomes more pronounced. Thus, the temperature increase will result in rather counterintuitive changes in the diffraction data as positions of nitrogen will be better defined at a higher temperature than near 0 K. The second and higher excited vibrational levels of DyGd 2 N@C 80 resemble the levels of a system with a single energy minimum. A temperature increase above 40 K leads to a gradual shift of the probability maxima to higher |z| values and broadening of the probability density. The decrease of the cluster size to that of Dy 3 N results in a conceptual change of the form of the potential energy profile. Although its double-well nature with the minima near |z| = 0.2 Å is formally preserved, the inversion barrier is reduced to a small value of 1.5 cm −1 , which is already comparable to the numerical precision of the computational approach. The ground vibrational level is found at 25.3 cm −1 , considerably higher than the inversion barrier, and |Ψ 0 2 | has a single maximum at z = 0. Essentially the structure of vibrational levels in Dy 3 N@C 80 is that of a system with a single well potential. Substantial anharmonicity of the potential, which has ∼z 4 dependence, leads to a non-equidistant spacing of the levels. The probability density for the nitrogen position in Dy 3 N@C 80 has a single-peak shape at all temperatures. The peak is rather broad and gets flattened at higher temperatures. Thus, one can expect that the diffraction data for Dy 3 N should show a single nitrogen position with an anomalously elongated thermal ellipsoid as was indeed reported in ref. 31.
A situation similar to Dy 3 N is also predicted for the DyEr 2 N and Lu 3 N clusters. The pyramidalization of the cluster and inversion barrier in DyEr 2 N are even smaller than those in Dy 3 N. In Lu 3 N@C 80 , a single energy minimum is found. The potential is not harmonic even for the relatively small Lu 3 N cluster and for both molecules it is well described by a quartic function with only a small contribution of z 2 . A decrease of the cluster size results in a narrowing of the potential well and an increase of the distance between vibrational levels. Accordingly, the probability density with the maximum at z = 0 becomes narrower with the decrease of the cluster size. Thus, diffraction data should give a planar structure of the M 3 N cluster with a gradual decrease of the elongation of the nitrogen ellipsoid from Dy 3 N over DyEr 2 N to Lu 3 N. Fig. 11 compares probability densities, obtained from computational studies of DyGd 2 N@C 80 and DyEr 2 N@C 80 , to the difference in the electron density near the position of the nitrogen, obtained by subtracting the density reconstructed from the refined positions of all atoms except nitrogen from the experimental density determined by X-ray diffraction. Good agreement between experiment and theory can be observed. For DyGd 2 N@C 80 , an elongated density profile with two well-defined maxima can be distinguished and the maxima are close to the maxima in the probability density function. For DyEr 2 N@C 80 , a single density lobe is seen and its spatial extension along the z-axis corresponds to the peak shape of the probability density.
To summarize, the analysis of the potential energy profile along the cluster pyramidalization coordinate shows that precise structure determination of the M 3 N cluster from diffraction data may be complicated. Calculations show a gradual development of the double-well potential with increasing metal size, which means that the pyramidalization of the M 3 N cluster should increase gradually as long as the metal size is larger than that of Lu. However, if the inversion barrier is below the lowest vibrational level, the probability density has a single maximum (Fig. 9b and 10). Diffraction methods (which do not restore the shape of the potential but determine highprobability atomic positions) should then give the planar shape of the cluster even when the energy profile has two minima. At the same time, the thermal ellipsoid of nitrogen should be strongly elongated in the direction perpendicular to the metal plane. Description with single or split nitrogen positions may then give similar R-factors. A true pyramidal shape of the cluster can be obtained when at least one vibrational level of the system lies below the inversion barrier. But even in this case, a mismatch between the shape of the potential and probability density function, i.e. when the maxima in the latter do not correspond to the minima in the former, can be rather strong, as found here for the DyGd 2 N cluster, and pyramidalization deduced from diffraction data should exhibit considerable temperature dependence. Only when two or more vibrational levels are below the inversion barrier, as found for Gd 3 N@C 80 , a stable pyramidal shape of the cluster with moderate temperature dependence can be observed by X-ray diffraction. Needless to say, additional disorder of the fullerene cage and metal positions will further complicate the situation. Thus, the molecular structure elucidation of endohedral fullerenes with labile endohedral species at a single temperature may give ambiguous results. Variable temperature studies with Fig. 11 Difference electron density maps for the nitrogen atom in DyEr 2 N@C 80 (top) and DyGd 2 N@C 80 (bottom). The calculated density was built using refined positions and structure factors of all atoms except nitrogen. 2D maps of the density are overlaid with the probability density function for the nitrogen atom at the z-coordinate computed for 100 K (the same as in Fig. 9). For the sake of comparison the color scale and the scale for probability density are the same for both molecules.
careful analysis of thermal motion are needed to address this problem. 43 In the end we would like to note that parameters of the double-well potentials found in this work for M 3 N@C 80 molecules with large endohedral clusters may be of use for design of the sources of terahertz radiation or even masers. The very first two-state maser system designed back in 1954 44 utilized ammonia gas and operated at a frequency of 24.0 GHz corresponding to an inversion splitting of 0.8 cm −1 in an NH 3 molecule. The inversion barrier of 177 cm −1 in Gd 3 N@C 80 is almost an order of magnitude smaller than the insertion barrier of 2020 cm −1 in NH 3 , but the vibration effective mass of 14 Da is an order of magnitude greater in Gd 3 N@C 80 against 1 Da for ammonia. Thus, the transmission coefficient for the quantum tunneling defined as exp À2 Ð Δz dz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m eff EðzÞ À ε j j p À Á in the Wentzel-Kramers-Brillouin approximation is of the same order for both molecules and hence similar quantum dynamics can be expected for them. Without going into deep detail at this moment, we just highlight that the Gd 3 N@C 80 system may be driven and modulated by an external electric field in the same way and fashion as is possible for the ammonia system and hence Gd 3 N@C 80 is a prospective candidate for a two-state maser system with an operating frequency of 60.0 GHz (inversion splitting: 2 cm −1 ). A combination of other lanthanides can be used to tune the shape of the double-well potential and hence the inversion splitting and operating frequency. Differently from the ammonia gas, in the case of the fullerene crystal the M 3 N units are encapsulated in carbon cages and hence are well-separated from each other and isolated from the environment. Thus, the M 3 N-based maser combines the simplicity of the ammonia maser (twostate system; no need for pumping as the energy split is small; a lack of spontaneous emission) with higher efficiency in amplification and higher photon densities due to the solid state density of the M 3 N units in the crystal. 45

Conclusions
In this work we have undertaken a systematic exploration of various Dy-lanthanide combinations in the synthesis of mixedmetal nitride clusterfullerenes. A variation of the size of the metal combined with Dy offered a possibility to gradually tune the size of the nitride cluster and study the influence of this factor on the synthesis and structure. The ionic radius of the metal employed in the synthesis is shown to be the main factor affecting the relative yield of the majority of Dy x M 3−x N@C 80 species. Yet, the electronic properties of metals such as the third ionization potential, also play an important role and may cause completely unexpected distribution of products, such as that demonstrated in this work for the Dy-Tm system.
A single-crystal X-ray diffraction study of the molecular structures of DyGd 2 N@C 80 and DyEr 2 N@C 80 along with DFT computation for a broad range of Dy x M 3−x N@C 80 molecules offered a possibility to analyze how the variation of the size of the nitride cluster affects its shape. DFT calculations showed that a gradual increase of the metal size leads to the variation of the cluster shape from a planar triangle when the cluster is relatively small to a trigonal pyramid with gradually increasing height when the metal size increases. For pyramidal clusters the energy profile along the pyramidalization coordinate has a double-well shape. At the same time, calculations show that the inversion barrier for the pyramidal inversion in the M 3 N cluster is quite small. This situation results in a fundamental problem for structure determination rooted in the quantum nature of molecular vibrations: if the energy of the zeroth vibrational level is higher than the inversion barrier, the apparent structure of the pyramidal M 3 N cluster as perceived from diffraction data will be planar. Particularly interesting is the situation when the inversion barrier is comparable to the lowest-energy vibrational levels. In this case, quantum effects may reveal themselves as the strong variation of the probability distribution with temperature and potentially may be detected by variable-temperature X-ray diffraction experiments or spectroscopic studies. Finally, we propose that the double-well potential and inversion splitting of the levels for larger M 3 N clusters have a number of parameters which make them prospective candidates for terahertz light applications.

Conflicts of interest
There are no conflicts to declare.