An atlas of endohedral Sc₂S cluster fullerenes

Abstract

Structural identification is a difficult task in the study of metallofullerenes, but understanding of the mechanism of formation of these structures is a pre-requisite for new high-yield synthetic methods. Here, systematic density functional theory calculations demonstrate that metal sulfide fullerenes Sc₂S@C_n have similar cage geometries from C₇₀ to C₈₄ and form a close-knit family of structures related by Endo–Kroto insertion/extrusion of C₂ units and Stone–Wales isomerization transformations. The stabilities predicted for favoured isomers by DFT calculations are in good agreement with available experimental observations, have implications for the formation of metallofullerenes, and will aid structural identification from within the combinatorially vast pool of conceivable isomers.

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1. Introduction

Endohedral metallofullerenes (EMFs) are closed cage-shaped molecules with encaged metal atoms or clusters.^1,2 Since their first synthesis in macroscopic amounts,³ EMFs have attracted extensive interest from chemists, physicists and materials scientists. To date, more than two hundred EMFs have been characterized, including mono-metallofullerenes, di-metallofullerenes, tri-metallofullerenes, and fullerenes incorporating clusters of various kinds.⁴ Amongst the EMFs, those in which the encapsuland is a metallic cluster have attracted attention because of their novel geometries and properties.^5,6 However, as far as we are aware, the mechanism of formation of EMFs has not yet been clarified, even though so many EMFs have been reported. For small fullerenes, a recent theoretical study shows that C₂ insertion (by the Endo–Kroto mechanism) can facilitate the formation of larger fullerenes without additional Stone–Wales (S–W) rotations.⁷ Where the cage is larger than C₇₀, direct C₂ insertion in classical isomers will form non-IPR (IPR = isolated pentagon rule⁸) isomers, which is not in agreement with experimental observations that all reported stable (neutral, bare) fullerenes are IPR-satisfying. Hence, C₂ insertion alone cannot account for the formation of fullerenes. Some post-insertion step, for which S–W rotation is the best candidate, is a necessary stage in the formation of fullerenes.

For endohedral metallic cluster fullerenes, formation processes may be similar to those for bare fullerenes, at least in terms of the cages involved, but they are likely to be affected by the presence of the additional participant. A pre-formed fullerene cage would almost certainly be broken by insertion of a metallic cluster into its interior, as the available windows, hexagonal, pentagonal or heptagonal faces, are small. It seems highly improbable that a metallic cluster will insert to form an EMF directly. Thus, reasonable formation paths would seem to require growth or shrinkage of a cage with a cluster inside, or curling up to form fullerene cages with simultaneous encapsulation of a metal atom/cluster, as is consistent with molecular dynamics simulations.⁹

Recent experiments show that non-classical fullerene cages with a heptagonal ring can also encapsulate metallic clusters,¹⁰ or can be captured as chlorofullerenes¹¹ from the carbon-arc plasma in situ. Theoretical studies show that heptagonal rings may play an important role in the formation of bare fullerenes^12,13 and trimetallic nitride template fullerenes.¹⁴ To predict the geometrical structures of some poorly characterized metallofullerenes and seek insight into their formation, it was decided to make a systematic study of classical and non-classical isomers Sc₂S@C_n (n = 70–84). This was carried out with the help of an extended face-spiral algorithm for construction of cage candidates with small numbers of heptagonal faces. The results show that heptagon-including metallic sulfide fullerenes Sc₂S@C_n are not competitive with classical Sc₂S@C_n in terms of total energy. Interestingly, there are strong structural similarities amongst low-energy Sc₂S@C_n isomers of equal and adjacent cage sizes. These similarities are given concrete form in terms of S–W isomerization and C₂ insertion/extrusion transformations. Our results provide clues to finding new metallofullerenes from within the tens of thousands of conceivable structural isomers. They also give potentially useful information on the formation mechanisms of EMFs.

2. Computational details

The isomers to be considered are generated by an extension of the face-spiral algorithm to allow up to one heptagonal and 13 pentagonal faces.¹⁵ This approach is known to be complete in the size range. The nomenclature and labelling of isomers follow the face-spiral algorithm. Briefly, classical isomers are labelled by their positions in the sequence of canonical spirals, and a superscript (1h) is used to indicate a non-classical isomer with one heptagon, with a number indicating its position in the spiral order for these cages. Topological coordinates¹⁵ are used to provide initial cage structures, which are then optimized for charges 0, −2, −4 and −6, first at the semi-empirical PM3 level and then, for a selection of the best cages at each charge, at the B3LYP/3-21G level. Based on the energy ranking for the optimized cages with charge −4, and some other favoured isomers with different charges, the favoured cages are used as parents to construct Sc₂S-based EMFs. The Sc₂S moiety is placed close to the geometric centre of the cage, and irrespective of the initial shape and atom ordering (Sc–S–Sc or S–Sc–Sc) optimizes to a structure where S lies between the two Sc centres; this is a simple consequence of the fact that the moiety donates electrons to the cage, generating electrostatic repulsion between the newly formed ionic Sc centres. The numbers of isomers considered by nuclearity are shown in S1 (ESI†). Geometrical optimizations are performed at B3LYP/3-21G and then B3LYP/6-31G* levels on Sc₂S@C_n with n from 70 to 84. All calculations are performed with Gaussian 09 software.¹⁶ The results for the favoured isomers are shown in Fig. 1 and Table 1, and the optimized coordinates calculated for the isomers of lowest energy are provided in S2 (ESI†). In the optimized structures for the favoured isomers, the Sc–S–Sc angle varies between 140° and 180° in most cases, with Sc₂S@C₇₆:19151 having a much smaller angle of 108°. The linear Sc–S–Sc moieties are found in Sc₂S@C₇₄:13333 and Sc₂S@C₈₄:51591 (see coordinates in S2, ESI†).

Fig. 1 Optimized structures of the three isomers of lowest energy for each Sc₂S@C_n with n from 70 to 84 as predicted at the B3LYP/6-31G* level.

Table 1 Numbers of pentagon adjacencies (N₅₅), relative energy (ΔE, kcal mol⁻¹), HOMO–LUMO gap (eV) and Sc–Sc distance (Å) for the lowest-energy isomers of Sc₂S@C_n

Cage	N 55	ΔE	Gap	Sc–Sc	Cage	N 55	ΔE	Gap	Sc–Sc	Cage	N 55	ΔE	Gap	Sc–Sc
C70:7892-C2	2	0.0	1.84	3.597	C76:19151-Td	0	0.0	1.45	3.801	C82:39715-Cs	0	0.0	1.72	4.073
C70:7957-C2	2	20.8	1.54	4.584	C76:17490-Cs	2	11.3	1.32	4.546	C82:39717-C3v	0	0.3	2.08	4.008
C70:7924-C1	2	20.9	1.35	4.241	C76:19138-C1	1	12.0	0.76	4.599	C^1h82:155199(C1)	1	12.8	1.81	4.129
C72:10528-Cs	2	0.0	1.85	4.225	C78:24088-C2v	2	0.0	1.76	4.207	C82:39718-C2v	0	14.4	1.20	4.363
C72:10616-Cs	2	13.9	1.93	4.425	C78:24109-D3h	0	1.7	0.97	4.485	C84:51591-D2d	0	0.0	1.68	4.621
C72:10530-C1	2	19.1	1.66	3.979	C78:24107-C2v	0	3.7	1.53	4.168	C84:51383-C1	1	10.4	1.90	4.447
C74:13333-C2	2	0.0	1.68	4.651	C80:31923-D5h	0	0.0	1.12	4.108	C84:51580-C1	0	12.3	1.73	4.163
C74:14246-D3h	0	11.8	1.14	3.815	C80:31922-C2v	0	9.0	1.67	3.840
C74:14239-C2v	2	11.9	1.21	4.157	C80:31924-Ih	0	9.2	0.84	4.196

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3. Results and discussion

Fig. 1 shows the structures for the three most favoured isomers for each of the eight formulas Sc₂S@C_n with n = 70–84.

It is shown in Table 1 that the isomer of lowest energy isomer for Sc₂S@C₇₀ has a large HOMO–LUMO gap (1.84 eV) and parent cage, the non-IPR C₇₀:7892, with two pentagon adjacencies. The stability of this isomer is in good agreement with experimental results.¹⁷ In fact, this cage has recently been shown to be the parent of a geometrically and electronically similar cluster Sc₂O.¹⁸ The two isomers next in energy order are also classical, but lie more than 20 kcal mol⁻¹ higher.

It is shown in Table 1 that Sc₂S@C₇₂:10528 is the first favoured isomer in energy terms and has a large gap of 1.85 eV. The next most favoured isomer is higher in energy by 13.9 kcal mol⁻¹ than the first, which corresponds to an equilibrium fractional population of only 3% at 2000 K. This isomer and others of higher energy are unlikely to be isolated in significant yield. The isomer predicted to be most favoured corresponds to experimental observations^19a and recent theoretical results.^19b The same parent cage is shared by the electronically similar cluster Sc₂C₂ in the metallic carbide fullerene Sc₂C₂@C₇₂,²⁰ and the electronically and geometrically similar cluster Sc₂O in the metallic oxide fullerene Sc₂O@C₇₂.²¹

We have also studied Sc₂S@C₇₄.²² The results show the most favoured isomer to be IPR-violating and Sc₂S cluster to be linear inside cage C₇₄:13333. The next two isomers in order of energy lie 11.8 and 11.9 kcal mol⁻¹ higher in energy than the first. The cage of the second favoured isomer is the only IPR isomer of C₇₄, which itself is open-shell and hence reactive as a neutral cage, but is apparently passivated in the form of Sc₂S@C₇₄.

A molecule of stoichiometry Sc₂S@C₇₆ has been detected by mass spectrometry, but without further experimental characterization.¹⁹ Our calculations indicate that the isomer of lowest energy is Sc₂S@C₇₆:19151, followed by Sc₂S@C₇₆:17490, in agreement with recent theoretical results.²³ The third isomer in order of energy is Sc₂S@C₇₆:19138. XRD characterization has shown that the C₇₆:19151 cage also encapsulates the Sc₂O cluster to form Sc₂O@C₇₆.²⁴

For Sc₂S@C₇₈, the isomer predicted to have lowest energy has cage C₇₈:24088, a non-IPR fullerene with two pentagon pairs, instead of the well-known IPR cage, C₇₈-24109, which encapsulates two La atoms in the form of La₂@C₇₈²⁵ in relatively high yield. Cage C₇₈:24088 would be only ninth best in terms of the energy of the tetra-anion C₇₈⁴⁻, reminding us that a simple electron-transfer model does not account for the stability of Sc₂S@C_n in every case.

The cage of the most favoured Sc₂S@C₈₀ is IPR-satisfying C₈₀:31923, which gives the C₈₀ tetra-anion of second lowest energy (see S3, ESI†). The tetra-anion with lowest calculated energy is based on I_h-symmetrical C₈₀:31924, but the triply degenerate LUMO of this cage would tend to indicate acceptance six additional electrons, rather than four that Sc₂S can offer, and it turns out that the isomer of Sc₂S@C₈₀ based on the I_h cage has only the third best predicted energy. The cage of the second favoured Sc₂S@C₈₀ is IPR-satisfying C₈₀-31922, which has recently been shown to be the parent of a geometrically and electronically similar cluster Sc₂O@C₈₀.²⁶ This case demonstrates some of the complexities of the interaction between cage and encapsuland, where predictions of stability based on electronic factors may be in conflict with simple steric matching of size and shape.

In terms of calculated relative energies of the tetra-anion (see S3, ESI†), C₈₂:39717 should be the best cage for encapsulating a Sc₂S cluster and, in fact, this metallic sulfide fullerene was the first to be reported for any nuclearity.²⁷ Our calculations predict that Sc₂S@C₈₂:39717 and the isomer of lowest calculated energy, Sc₂S@C₈₂:39715, are essentially iso-energetic. Although evidently smaller than that of Sc₂S@C₈₂:39717, the HOMO–LUMO gap of Sc₂S@C₈₂:39715 (1.71 eV) is still relatively large, and this isomer has recently been reported in experiment.²⁸ Interestingly, non-classical C^1h₈₂:155199, the cage of the third best isomer of Sc₂S@C₈₂ in our calculations, could serves as a structural bridge between the cages of two experimental isomers of Sc₂S@C₈₂, as shown in Fig. 2.

Fig. 2 The structural web (atlas) of Sc₂S@C_n isomers. Arrows in red and purple indicate that a Stone–Wales isomerization can interconvert two isomers; colored edges indicate the atoms undergoing a Stone–Wales rotation.

For Sc₂S@C₈₄, the calculations show that the favoured isomer is IPR-Sc₂S@C₈₄:51591, sharing a parent cage with Sc₂C₂@C₈₄;²⁹ the second lowest isomer is non-IPR Sc₂S@C₈₄:51383, with predicted high kinetic stability. Our recently reported calculations show that Sc₂S@C₈₄:51575 is favoured at high temperature (∼2800 K); this last isomer can transform to the most favourable isomer Sc₂S@C₈₄:51591 via S–W rotation.³⁰

Geometry optimizations on Sc₂S@C_n with non-classical candidate cages show none that are competitive with classical isomers. The best of the non-classical Sc₂S@C_n isomers with n = 70 to 82 lie respectively 28.5, 32.5, 34.9, 27.8, 34.3, 17.9 and 28.2 kcal mol⁻¹ higher than their classical competitors. Thus, isolation of non-classical isomers of these compounds is unlikely.

Cluster effect

I _h-C₈₀ is the commonest parent cage for encapsulation of TNT clusters, as demonstrated by numerous experimental observations.^4–6 However, it is not favoured for encaging Sc₂S. There are other indications of differences in stability for Sc₂S and TNT clusters. For example, as mentioned above, C₈₂:39718, the cage of the most favoured isomer of Sc₃N@C₈₂³¹ is a structural bridge between the cages of two favoured candidates, i.e., C₈₂:39717 and C₈₂:39715 for encaging TNT clusters. However, when the encaged cluster is Sc₂S, the bridging isomer is the non-classical isomer C^1h₈₂:155199. Significantly, C₂ addition to C₈₂:39715, the favoured cage for encaging Sc₂S, can form C₈₄:51383, the parent cage of the second Sc₂S@C₈₄; however, C₂ addition to C₈₂:39718, the parent cage of the favoured Sc₃N@C₈₂ can form C₈₄:51365, the parent cage of EMFs M₃N@C₈₄ (M = Tb, Tm and Gd).^32,33 These results suggest that the encaged cluster mediates, or even controls, the formation mechanism; this important role could arise from differences in electron-donating capacity (six electrons for TNT cluster and four for the Sc₂S or Sc₂O cluster) and/or different degrees of geometrical match to the cages.

Our recent calculations have shown that the lower symmetry and local deformations associated with introduction of a heptagonal ring favour encapsulation of mixed (and intrinsically less symmetrical) metal nitride clusters.¹⁴ In the present case, however, no non-classical Sc₂S-based EMF is predicted to be competitive with those based on classical cages. Non-classical cages that include one heptagonal face will also tend to have more pentagon adjacencies, almost universally³⁴ destabilizing in the neutral. As there are two metallic atoms rather than three, it may be easier for them to find suitable internal positions, whether the cage is symmetrical or not, especially as Sc₂S has a low bending force constant^35,36 and hence is intrinsically more flexible than an M₃N cluster. This may account for the lesser role of non-classical cages in encapsulation of the Sc₂S cluster.

Structural interdependence

The calculations here give a set of favoured structures for Sc₂S-based EMFs with eight different cage sizes. As noted above, all predictions for isomers of lowest energy are in agreement with available experimental and theoretical data. Interestingly, there is an evident structural dependence among the parent cages of the low-energy isomers Sc₂S@C_n. As shown in Fig. 2, a C₂ unit added to C₇₀:7892 can form C₇₂:10528, and a further C₂ addition forms C₇₄:13333. The corresponding Sc₂S-based EMFs are either those experimentally reported or theoretically predicted to be of lowest energy. A further addition leads to C₇₆:17490, the parent cage of the second favoured isomer of Sc₂S@C₇₆. Further C₂ addition to C₇₆:17490 can form C₇₈:22010, the parent cage of many TNT fullerenes;^37,38 another C₂ addition can form C^1h₈₀:112912, one Stone–Wales step away from the well-known I_h-symmetric C₈₀:31924, the favoured cage for hexavalent (M₃N) clusters. Carrying on the process of C₂ addition can lead to the parent cage (C₈₂:39717) of the first experimentally reported isomer of Sc₂S@C₈₂, and the parent cage of the recently reported, and essentially iso-energetic isomer, Sc₂@C₈₂:39715 with consecutive S–W rotations. C₂ addition on C₈₂: 39718 can form C₈₄:51365, the parent of M₃N@C₈₄ (M = Tb, Tm and Gd).^32,33

In another part of the map, the unique IPR isomer of C₇₀, i.e., C₇₀:8149, has a tetra-anion energy that makes it the favoured candidate for encaging a cluster that would donate four electrons. C₂ addition to C₇₀:8149 can lead to C₇₂:11188, fifth best in terms of its tetra-anion energy. In turn, C₇₂:11188 can isomerize into a non-classical isomer C^1h₇₂:29907 with two pairs of fused pentagons, which can form C₇₄:14246 via directly C₂ addition. C₇₄:14246 can form C₇₆:19138 and C₇₈:24088 via successive C₂ addition. Interestingly, C₂ addition on the parent cage of most favoured Sc₂S@C₈₀:31923 can lead to C₈₂:39663, which has recently been proved to be the parent of Gd₃N@C₈₂,^39,40 this cage is an S–W rotation away from the parent cage of the most favoured isomer Sc₂S@C₈₂:39715.

Exploration of the details encoded in the figure show that the vertical and diagonal connections in the map provide a route for expansion/shrinkage of all eight parent cages of low-energy isomers from Sc₂S@C₇₀ to Sc₂S@C₈₄.

C₇₀:7892, the parent of the most favoured isomer of Sc₂S@C₇₀ can transform into C₇₀:8111 via a C₂ extrusion (C₆₈:6094) and C₂ insertion, and then isomerize into C₇₀:8149; C₆₈:6094 has been captured as C₆₈Cl₈ by in situ chlorination in the gas phase during radio-frequency synthesis⁴¹ and theoretically predicted to be the parent of Sc₂O₂.⁴² C₇₂:10528, the parent of the most favoured isomer of Sc₂S@C₇₂, has a transformation to C₇₂:11188, one of the molecules of lowest energy for this stoichiometry. The isomer of lowest energy Sc₂S@C₇₄:13333, can transform into the second-best Sc₂S@C₇₄:14246 as shown in Fig. 2. C₇₆:14790 can transform into C₇₆:19138, the second best cage for both tetra-valent and di-valent clusters, according to the relative energies of the anions. In fact, C₇₆:19138 has been shown to encage divalent Sm.⁴³ C₇₆:19142 can transform into C₇₆:19151 via an S–W rotation (not shown), the parent cage of lowest energy for Sc₂S@C₇₆. C₇₈:24099 can transform into C₇₈:24109 via an S–W rotation, the isomer most favoured for encaging Sc₂O.⁴⁴ C₇₈-22010, the most favoured cage for the large metallic cluster Gd₃N³⁷ can transform into C₇₈-24088, the most favoured cage for Sc₂S. Interestingly, C₇₀:7892, C₇₂:10528, C₇₄:13333, C₇₆:17490 and C₇₈:22010 all have similar transformations (extrusion/insertion and then isomerization) to their low energy counterparts. The I_h-C₈₀ cage can transfer into D_5h-C₈₀via three kinds of path; the first is by growth, shrinkage and isomerization, the second is by shrinkage, growth and isomerization; the third path (at least in in a formal mathematical sense) is by a 36° rotation around a C₅ axis of one half of I_h-C₈₀ against the other. C₈₂:39717 can isomerize into C₈₂:39715, the parent cage of a recently experimentally reported isomer Sc₂S@C₈₂²⁸via a non-classical bridge cage, C^1h₈₂-155199. Successive S–W rotations can transfer C₈₄:51546 into the most favoured neutral cage C₈₄:51590, and the parent cage of lowest energy Sc₂S@C₈₄, C₈₄:51591.

In summary, favoured cages can grow or shrink in the vertical direction on the map, or isomerize in the horizontal direction into other favoured cages. Horizontal connections in the map typically provide a possible route from one favoured isomer to another of Sc₂S@C_n. These include the global rotation of one hemisphere against another, which would have a high energy barrier for a pre-formed cluster, but for a mechanism involving combination of fragments would be a low-energy process. The parent fullerene cages and the low-energy Sc₂S-based EMFs form a web of complex genealogical relationships. Many of the cages in the near region of this map are favoured for encapsulation of other types of mono-metallic, di-metallic or metallic clusters, all demonstrating common stabilizing substructures and motifs.

Formation mechanisms

The calculations of optimal structures demonstrate that C₂ insertion/extrusion is not only a topological requirement but also a possible bridge for growth of favoured Sc₂S-based EMFs, connecting isomers of low energy. As the most common metallic atom(s) or clusters would have extremely high barriers to insertion through a pentagon or hexagon of the fullerene, it seems likely that the Sc₂S cluster would be encaged early in the process inside a fullerene (or fullerene-like) cage that could continue to grow or shrink via C₂ insertion or extrusion, eventually forming a stable EMF. D_3h-C₇₈, I_h-C₈₀, D_5h-C₈₀, C_2v-C₈₂, C_3v-C₈₂ and D_2d-C₈₄ are all IPR-satisfying and comprise the set of parent cages of most EMFs, but they cannot be formed by direct C₂ insertion into a lower IPR fullerene, and so at least one S–W isomerization step is necessary. This remains true in models of growth from smaller clusters, since even if an IPR-satisfying cage of C_n+2 can be formed directly via C₂ insertion into the heptagon of a non-classical isomer, an S–W isomerization would be needed for initial transformation from a classical isomer.

Molecular dynamics simulations have shown that hot giant fullerenes can lose or gain carbon in high temperature conditions.⁴⁵ This existence of a structural web such as shown in Fig. 2 suggests that, irrespective of the formation mechanism of Sc₂S@C_n, there are generally many Sc₂S@C_n species with different sizes, since any as-formed Sc₂S@C_n can produce other metallic sulfide fullerenes via top-down⁴⁶ or bottom-up⁴⁷ paths and S–W isomerization. Experiment shows that there are always many Sc₂S@C_n species in the soot,^17,19a,48 and multiple species with different metallic atoms or clusters encaged inside different fullerene cages are produced under the same reaction conditions. For example, a mixture of several Sc₂O@C_n, multiple Sc₂C₂@C_n, and even Sc₃N@C₈₀ species are found in soot produced under conditions designed to yield Sc₂O@C_n by introducing CO₂ as the oxygen source during the arcing process;²¹ a mixture of Sc₂S@C_n and Sc₂C₂@C₈₀ isomers is found in soot in a conventional Krätschmer–Huffman reactor for producing metallic sulfide fullerenes under an atmosphere of SO₂.¹⁹ The difficulty in isolating a particular EMF is mainly a result of the diversity of EMFs in reaction mixtures.

Recently, Wang et al. identified four key structural motifs which govern the relative energies of anions of classical fullerene cages and thus can be used to search for suitable cages in endohedral metallofullerenes.⁴⁹ This is a significant finding, but more rules would be desirable, since these motifs are compatible with many structural isomers. Our study helps to refine the predictive picture by setting the mass of experimental results scattered in the literature into the context of a relatively small family of low-energy isomers.

A further remark can be made. We have considered fullerenes encapsulating metallic sulfide clusters from the point of view of their structural interdependence. However, our main findings are likely to be valid for Sc₂C₂ and Sc₂O in fullerene cages. The carbide Sc₂C₂ has similar electronic properties to Sc₂S, and Sc₂O and Sc₂S are similar in both electronic and geometrical requirements. Thus, the present calculations may have implications for a larger family of EMFs, beyond the compounds of Sc₂S.

4. Conclusion

Systematic DFT calculations demonstrate that there is a close structural interdependence amongst the favoured isomers of Sc₂S@C_n with size from C₇₀ to C₈₄, and that one-heptagon non-classical Sc₂S@C_n are generally much less competitive relative to classical cages in terms of total energetics. This is quite unlike the situation that we found for tri-metallic nitride fullerenes. These results indicate that formation of Sc₂S@C_n and other EMFs is guided by the encaged clusters, and suggests a similar formation route in that cages featuring as preferred have closely related structures.

As a whole, the stability of EMFs is determined by two main factors: one is electron transfer and the other is effect of size. Calculations show that many of the lowest energy isomers of Sc₂O@C_n share the same cages with Sc₂S@C_n apart from the cases with n = 74, 78 and 84. Since both Sc₂O and Sc₂S tend to donate 4 electrons to a fullerene cage, they tend to select the same isomer as host cage and these results indicate that electron transfer interactions play a vital role. Of course, the cluster sizes are different, and rankings are evidently different when comparing structures involving a series of cages of size C_n (when the sizes and shapes of both cages and encaged clusters play an important role).

Acknowledgements

The authors thank Gunnar Brinkmann (Ghent University) for cross-checking completeness of the face-spiral construction for the non-classical cages discussed here. Li-Hua Gan thanks the National Natural Science Foundation of China (51272216, 51472208) for financial support.

References

1. H. Shinohara, Rep. Prog. Phys., 2000, 63, 843–892.
2. A. Rodríguez-Fortea, A. L. Balch and J. M. Poblet, Chem. Soc. Rev., 2011, 40, 3551–3563.
3. Y. Chai, T. Guo, C. Jin, R. E. Haufler, L. P. F. Chibante, J. Fure, L. Wang, J. M. Alford and R. E. Smalley, J. Phys. Chem., 1991, 95, 7564–7568.
4. A. A. Popov, S. Yang and L. Dunsch, Chem. Rev., 2013, 113, 5989–6113.
5. J. Zhang, S. Stevenson and H. C. Dorn, Acc. Chem. Res., 2013, 46, 1548–1557.
6. T. Wang and C. Wang, Acc. Chem. Res., 2014, 47, 450–458.
7. J. S. Dang, W. W. Wang, J. J. Zheng, X. Zhao, E. Osawa and S. Nagase, J. Phys. Chem. C, 2012, 116, 16233–16239.
8. H. W. Kroto, Nature, 1987, 329, 529–531.
9. Q. Deng, T. Heine, S. Irle and A. A. Popov, Nanoscale, 2016, 8, 3796–3808.
10. Y. Zhang, K. B. Ghiassi, Q. Deng, N. A. Samoylova, M. M. Olmstead, A. L. Balch and A. A. Popov, Angew. Chem., Int. Ed., 2015, 54, 495–499.
11. Y. Z. Tan, R. T. Chen, Z. J. Liao, J. Li, F. Zhu, X. Lu, S. Y. Xie, J. Li, R. B. Huang and L. S. Zheng, Nat. Commun., 2011, 2, 1–6.
12. E. Hernandez, P. Ordejon and H. Terrones, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63, 193403.
13. (a) W. W. Wang, J. S. Dang, J. J. Zheng and X. Zhao, J. Phys. Chem. C, 2012, 116, 17288–17293; (b) W. W. Wang, J. S. Dang, J. J. Zheng, X. Zhao and S. Nagase, J. Phys. Chem. C, 2013, 117, 2349–2357.
14. L. H. Gan, D. Lei and P. W. Fowler, J. Comput. Chem., 2016, 37, 1907–1913.
15. P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995.
16. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox. Gaussian 09; Revision A. 02, Gaussian Inc., Pittsburgh, PA, 2009.
17. N. Chen, M. Mulet-Gas, Y. Y. Li, R. E. Stene, C. W. Atherton, A. Rodríguez-Fortea, J. M. Poblet and L. Echegoyen, Chem. Sci., 2013, 4, 180–186.
18. L. Feng, M. Zhang, Y. Hao, Q. Tang, N. Chen, Z. Slanina and F. Uhlík, Dalton Trans., 2016, 45, 8142–8148.
19. (a) N. Chen, C. M. Beavers, M. Mulet-Gas, A. Rodríguez-Fortea, E. J. Munoz, Y. Y. Li, M. M. Olmstead, A. L. Balch, J. M. Poblet and L. Echegoyen, J. Am. Chem. Soc., 2012, 134, 7851–7860; (b) L.-J. Wang, R.-L. Zhong, S.-L. Sun, H.-L. Xu, X.-M. Pan and Z.-M. Su, Dalton Trans., 2014, 43, 9655.
20. Y. Feng, T. Wang, J. Wu, L. Feng, J. Xiang, Y. Ma, Z. Zhang, L. Jiang, C. Shu and C. Wang, Nanoscale, 2013, 5, 6704–6707.
21. M. Zhang, Y. Hao, X. Li, L. Feng, T. Yang, Y. Wan, N. Chen, Z. Slanina, F. Uhlík and H. Cong, J. Phys. Chem. C, 2014, 118, 28883–28889.
22. L. H. Gan, Q. Chang, C. Zhao and C. R. Wang, Chem. Phys. Lett., 2013, 570, 121–124.
23. P. Zhao, T. Yang, Y. J. Guo, J. S. Dang, X. Zhao and S. Nagase, J. Comput. Chem., 2014, 35, 1657–1663.
24. T. Yang, Y. Hao, L. Abella, Q. Tang, X. Li, Y. Wan, A. Rodríguez-Fortea, J. M. Poblet, L. Feng and N. Chen, Chem. – Eur. J., 2015, 21, 11110–11117.
25. B. Cao, T. Wakahara, T. Tsuchiya, M. Kondo, Y. Maeda, G. M. A. Rahman, T. Akasaka, K. Kobayashi, S. Nagase and K. Yamamoto, J. Am. Chem. Soc., 2004, 126, 9164–9165.
26. Q. Tang, L. Abella, Y. Hao, X. Li, Y. Wan, A. Rodríguez-Fortea, J. M. Poblet, L. Feng and N. Chen, Inorg. Chem., 2015, 54, 9845–9852.
27. L. Dunsch, S. Yang, L. Zhang, A. Svitova, S. Oswald and A. A. Popov, J. Am. Chem. Soc., 2010, 132, 5413–5421.
28. B. Q. Mercado, N. Chen, A. Rodríguez-Fortea, M. A. Mackey, S. Stevenson, L. Echegoyen, J. M. Poblet, M. M. Olmstead and A. L. Balch, J. Am. Chem. Soc., 2011, 133, 6752–6760.
29. C. R. Wang, T. Kai, T. Tomiyama, T. Yoshida, Y. Kobayashi, E. Nishibori, M. Takata, M. Sakata and H. Shinohara, Angew. Chem., Int. Ed., 2001, 40, 397–399.
30. C. Zhao, D. Lei, L. H. Gan, Z. X. Zhang and C. R. Wang, ChemPhysChem, 2014, 15, 2780–2784.
31. T. Wei, S. Wang, F. Liu, Y. Tan, X. Zhu, S. Xie and S. Yang, J. Am. Chem. Soc., 2015, 137, 3119–3123.
32. C. M. Beavers, T. Zuo, J. C. Duchamp, K. Harich, H. C. Dorn, M. M. Olmstead and A. L. Balch, J. Am. Chem. Soc., 2006, 128, 11352–11353.
33. T. Zuo, K. Walker, M. M. Olmstead, F. Melin, B. C. Holloway, L. Echegoyen, H. C. Dorn, M. N. Chaur, C. J. Chancellor, C. M. Beavers, A. L. Balch and A. J. Athans, Chem. Commun., 2008, 1067–1069.
34. A. S. Matias, R. W. A. Havenith, M. Alcami and A. Ceulemans, Phys. Chem. Chem. Phys., 2016, 18, 11653–11660.
35. Q. Deng and A. A. Popov, J. Am. Chem. Soc., 2014, 136, 4257–4264.
36. Q. Tang, L. Abella, Y. Hao, X. Li, Y. Wan, A. Rodríguez-Fortea, J. M. Poblet, L. Feng and N. Chen, Inorg. Chem., 2016, 55, 1926–1933.
37. C. M. Beavers, M. N. Chaur, M. M. Oimstead, L. Echegoyen and A. L. Balch, J. Am. Chem. Soc., 2009, 131, 11519–11524.
38. A. A. Popov, M. Krause, S. Yang, J. Wong and L. Dunsch, J. Phys. Chem. B, 2007, 111, 3363–3369.
39. M. Mulet-Gas, A. Rodríguez-Fortea, L. Echegoyen and J. M. Poblet, Inorg. Chem., 2013, 52, 1954–1959.
40. B. Q. Mercado, C. M. Beavers, M. M. Olmstead, M. N. Chaur, K. Walker, B. C. Holloway, L. Echegoyen and A. L. Balch, J. Am. Chem. Soc., 2008, 130, 7854–7855.
41. K. Yu. Amsharov, K. Ziegler, A. Mueller and M. Jansen, Chem. – Eur. J., 2012, 18, 9289–9293.
42. L.-H. Gan, D. Lei and C. Zhao, RSC Adv., 2015, 5, 30409–30415.
43. Y. Hao, L. Feng, W. Xu, Z. Gu, Z. Hu, Z. Shi, Z. Slanina and F. Uhlík, Inorg. Chem., 2015, 54, 4243–4248.
44. P. Zhao, M. Y. Li, Y. J. Guo, R. S. Zhao and X. Zhao, Inorg. Chem., 2016, 55, 2220–2226.
45. B. Saha, S. Irle and K. Morokuma, J. Phys. Chem. C, 2011, 115, 22707–22716.
46. J. Zhang, F. L. Bowles, D. W. Bearden, W. K. Ray, T. Fuhrer, Y. Ye, C. Dixon, K. Harich, R. F. Helm, M. M. Olmstead, A. L. Balsh and H. C. Dorn, Nat. Chem., 2013, 5, 880–885.
47. P. W. Dunk, M. Mulet-Gas, Y. Nakanishi, N. K. Kaiser, A. Rodríguez-Fortea, H. Shinohara, J. M. Poblet, A. G. Marshall and H. W. Kroto, Nat. Commun., 2014, 5, 1–8.
48. N. Chen, M. N. Chaur, C. Moore, J. R. Pinzón, R. Valencia, A. Rodríguez-Fortea, J. M. Poblet and L. Echegoyen, Chem. Commun., 2010, 46, 4818–4820.
49. Y. Wang, S. Díaz-Tendero, F. Martín and M. Alcamí, J. Am. Chem. Soc., 2016, 138, 1551–1560.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6cp07370k

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