Dynamic motion of an Lu pair inside a C₇₆(T_d) cage

Abstract

Relativistic density functional theory (DFT) computations were performed to investigate the dynamic motion of an encapsulated Lu pair inside a C₇₆(T_d) cage. The results revealed that the lowest-energy configuration of Lu₂@C₇₆(T_d) adopts C₂ symmetry; four electrons are transferred to the outer carbon cage and the two encapsulated Lu atoms form a metal–metal single bond (with an electronic structure of Lu₂⁴⁺@C₇₆⁴⁻), and the good electron delocalization in the C₇₆⁴⁻(T_d) cage partially contributes the thermodynamic preference of Lu₂@C₇₆(T_d). The rather small barrier (3.2 kcal mol⁻¹) for Lu₂ atoms to hop from one stable site to another leads to flexible motion of the Lu pair inside the parent fullerene cage, and the O_h symmetrical motion trajectory of two Lu atoms is consistent with the STM image. The computed ¹³C NMR spectrum with this trajectory also agrees well with the experimental results.

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Introduction

Endohedral metallofullerenes (EMFs) have rather unique properties unexpected in empty fullerenes.^1–3 These properties lead to the wide applications of EMFs in various fields, such as materials science, energy, medicine, etc.^4–11 Especially, the dynamic behavior of encapsulated atoms in the cage makes EMFs potential candidates for molecular and nano-devices.¹² Not surprisingly, many experimental and theoretical studies have been carried out to investigate the dynamic motion of encapsulated atoms in EMFs.¹³ The scenario of La or Y atom moving inside C₈₂ (with C₂ symmetry) was firstly predicted by Andreoni and Curioni using ab initio molecular dynamics (MD) simulations.¹⁴ Nishibori et al. provided the first experimental evidence of the La atom movement inside C₈₂ cage, their maximum entropy method (MEM)/Rietveld analysis using synchrotron powder diffraction data indicated that La atom has a giant bowl-shaped movement at room temperature.¹⁵ However, the DFT studies by Jin et al. revealed a conflicting picture,¹⁶ they suggested that La atom undergoes a boat-shaped motion with a small amplitude inside C₈₂(C_2v) fullerene cage, rather than the bowl or hemisphere model with a large amplitude obtained from MEM/Rietveld analysis.

M₂@C₈₀ (M = La, Ce) are the representatives of dimetallofullerenes. For La₂@C₈₀(I_h), two La atoms circulate three-dimensionally inside the C₈₀(I_h) cage, which was supported by the two carbon signals in ¹³C NMR spectrum and one peak in ¹³⁹La NMR spectrum in experiments.¹⁷ A deeper insight into the La pair motion was given by MEM/Rietveld analysis, which demonstrated that the two La atoms present a trajectory that connects the six-membered rings of C₈₀(I_h), or a pentagonal-dodecahedron,¹⁸ as well as by DFT computations.¹⁹ For Ce₂@C₈₀(I_h), its ¹³C NMR spectrum indicated that the Ce atoms circulate randomly, as La atoms do in La₂@C₈₀(I_h).²⁰ The other isomer of Ce₂@C₈₀, which has a D_5h C₈₀ outer cage, was synthesized by Yamada et al.,²¹ the ¹³C NMR analysis and DFT computations revealed that the Ce atoms circulate two-dimensionally along a band of 10 contiguous hexagons inside the C₈₀(D_5h) cage.

C₇₆ fullerene has two structural isomers (D₂ and T_d) satisfying the isolated pentagon rule (IPR). The pristine C₇₆(T_d) isomer has not been synthesized yet due to its inherent open shell electronic structure. Excitingly, in 2010, Umemoto et al. synthesized and isolated Lu₂@C₇₆ for the first time.²² The UV-Vis-NIR, STM and ¹³C NMR chemical shift analysis disclosed that the entire Lu₂@C₇₆ molecule has T_d symmetry, and the charge state of (Lu₂)⁶⁺@C₇₆⁶⁻(T_d) was suggested. They also speculated that the two encapsulated Lu atoms rapidly rotate along a rhombic dodecahedron inside the cage to maintain the high symmetry. However, in 2011, by means of systematic density functional theory (DFT) computations and statistical thermodynamic analysis, Yang et al.²³ revealed that the inner two Lu atoms form a metal–metal single bond, and only transfer four electrons to the fullerene sphere, resulting in a formal valence state of (Lu₂)⁴⁺@C₇₆⁴⁻(T_d); they also suggested that the Lu₂ dimer can hop rapidly between six equivalent configurations in the fullerene cage at room temperature, giving rise to a trajectory as a tetrahedron in C₇₆(T_d). Such a controversy caught our interest: what is the most stable configuration for Lu₂@C₇₆? What is its bonding nature? What is the energy barrier for the dynamic motion of Lu atoms? Is the motion scenario a rhombic dodecahedron or a tetrahedron?²²

In this paper, by means of the relativistic DFT computations, we first investigated different isomers of Lu₂@C₇₆ and located the transition states for the Lu motion between the lowest-energy isomers. The bonding nature of the thermodynamically most favorable Lu₂@C₇₆(T_d) isomer was analyzed by quantum theory of atoms in molecules (QTAIM) method. Our computations support Yang et al.'s recent electronic structure assignment (Lu₂⁴⁺@C₇₆⁴⁻). Then, by analyzing the partial trajectory of Lu atoms surrounding the molecular geometric center, we obtained the O_h symmetrical motion trajectory of two Lu atoms inside C₇₆ cage. Furthermore, following the Lu atoms motion trajectory, we calculated the ¹³C NMR spectrum of Lu₂@C₇₆. Both the computed Lu motion trajectory and ¹³C NMR spectrum agree well with the experimental measurement.

Computational methods

ADF2008.01 program^24–26 and DMol³ code²⁷ were employed for the relativistic DFT calculations. The relativistic effects were taken into account by using the zero-order regular approximation (ZORA)^28–30 basis sets in ADF2008.01, as well as all electron relativistic core treatment in DMol³.

By ADF2008.01 program, full geometry optimizations without symmetry constraints were carried out using the generalized gradient approximation (GGA)³¹ exchange–correlation functional BLYP^32,33 with a triple-zeta polarized (TZP) basis set (referred to as GGA-BLYP/TZP). In addition, the frozen-core approximation up to the 1s orbital for C atoms and the 5p orbital for Lu atoms was used. By DMol³ code, geometry optimizations and transition state calculations were done at the GGA-BLYP/DNP level.

To disclose the bonding nature of Lu₂@C₇₆, we performed QTAIM analysis^34,35 of the most stable Lu₂@C₇₆. The wavefunction was generated at BLYP/6-31G* ∼ SDD³⁶ in Gaussian using the optimized geometries (BLYP/TZP in ADF). We also tested BLYP/6-31G* ∼ SARC³⁷ and BP86/6-31G* ∼ SARC^32,33,38 methods, and they gave very close results to that of BLYP/6-31G* ∼ SDD.

To better understand the inferred extraordinary electronic structure of Lu₂⁴⁺@C₇₆⁴⁻, we evaluated the electron delocalization (aromaticity) of the low-lying C₇₆⁴⁻ and C₇₆⁶⁻ isomers. Nucleus-independent chemical shifts (NICS, in ppm), a simple and efficient method to evaluate aromaticity,^39,40 were computed at the cage centers of the optimized geometries of the empty cages with Gaussian 09 program.⁴¹

The ¹³C NMR spectrum was calculated using gauge-independent atomic orbital (GIAO) method⁴² at the GGA-BLYP/TZP theoretical level by using the ADF2008.01 program. The chemical shifts were first evaluated relative to C₆₀, then were referenced to the carbon disulfide (CS₂) (δ(C₆₀) 143.15 ppm vs. CS₂).⁴³

Results and discussions

Searching for the lowest-energy isomer and its bonding nature

To clearly describe the symmetry of a C₇₆(T_d) cage, we take a triangle patch under a circum-spherical surface as a representative patch of the C₇₆(T_d) cage (Fig. 1). The area of this patch is equal to 1/24 of the total C₇₆ surface. On this representative patch, 16 key points are located at five kinds of carbon atoms (points C1, C2, C3, C4, C5), at seven kinds of bond center (point b1, b2, b3, b4, b5, b6, b7), at one five-membered ring's center (point p), and at three six-membered rings' centers (point h1, h2, h3). Every key point has its own local symmetry. For example, point L is at the intersectional carbon atom of three six-membered rings with C_3v local symmetry; point N is at the center of the six-membered ring with C_3v local symmetry; point M is at the center of 6–6 bond with C_2v local symmetry. We define point O as the geometric center of the carbon cage. Then the boundary of the patch can be characterized by a threefold axis OL, a two-fold axis OM and a three-fold axis ON. Note that all the independent symmetrical elements belonging to the T_d point group can be found in the representative patch. The entire surface of the polyhedron can be encompassed by performing different elemental symmetry operations on the patch. Therefore, we can consider the patch LMN (shaded) as the smallest structural unit rather than the whole cage.

Fig. 1 (a) The geometric structure of C₇₆(T_d) cage with the representative triangle LMN path; (b) all the key points on the patch: C1, b7 and h2 represent a three-fold axis, a two-fold axis, and a three-fold axis, respectively.

When the two Lu atoms are encapsulated in the cage, the system reduces from the original T_d symmetry of C₇₆ to a lower symmetry depending on the relative positions of the two Lu atoms inside the cage. Corresponding to the key points in the representative patch, full geometry optimizations were carried out for all 16 possible isomers, which resulted in five configurations, referred to as C₂, C₁, C_s, C_3v and D_2d according to the geometric symmetry. Table 1 summarizes the Lu–Lu distances (R_Lu–Lu), the shortest Lu–C distances (R_Lu–C), the relative energies and HOMO–LUMO gap energies for these five configurations of Lu₂@C₇₆. The results obtained by both ADF and DMol³ are presented.

Table 1 Key geometric parameters, relative energies and HOMO–LUMO gap energies for the five configurations of Lu₂@C₇₆. The results outside and inside the parentheses are from ADF and DMol³, respectively

Isomer	RLu–Lu (Å)	RLu–C (Å)	Relative energy (kcal mol^−1)	Gap (eV)
C2	3.36 (3.38)	2.42 (2.40)	0.0 (0.0)	0.75 (0.80)
C1	3.40 (3.46)	2.48 (2.43)	1.8 (0.4)	0.74 (0.79)
Cs	3.45 (3.50)	2.49 (2.45)	5.4 (3.2)	0.70 (0.75)
C3v	3.47 (3.50)	2.46 (2.42)	8.1 (4.7)	0.63 (0.72)
D2d	3.36 (4.00)	2.44 (2.41)	25.5 (24.8)	0.49 (0.60)

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ADF and DMol³ give the same order of relative energies and HOMO–LUMO gaps for the five isomers. The C₂ configuration is of the lowest energy and largest HOMO–LUMO gap. The thermodynamic stability is followed by the C₁ and C_s isomers. The more symmetrical isomers, C_3v and D_2d, have rather high relative energies and smaller HOMO–LUMO gaps.

Fig. 2 presents the top and side view of the C₂ configuration, the lowest energy isomer of Lu₂@C₇₆. Though the R_Lu–Lu and R_Lu–C of the five isomers are pretty close (see Table 1), the C₂ configuration has the smallest R_Lu–Lu and R_Lu–C (3.36 and 2.42 Å, respectively, by ADF; the corresponding values are 3.38 and 2.40 Å by DMol³).

Fig. 2 Top (a) and side (b) views of the lowest-energy isomer (with C₂ symmetry) of Lu₂@C₇₆. Lu atoms are green balls and C atoms are gray.

In order to confirm the reliability of the predicted relative stabilities, especially the frozen-core approximation for the basis sets, we carried out additional optimizations for the C₂ and C_3v isomers using three different function, namely GGA-BP86, PBE⁴⁴ and PW91,⁴⁵ together with the TZP all electron basis set using ADF package. The computational results summarized in Table 2 show that these function give rather similar optimized geometries and relative energies with those at GGA-BLYP/TZP. The R_Lu–Lu and R_Lu–C have almost no change, the biggest differences are within 0.03 Å. The relative energy difference between the two isomers is nearly the same, the C₂ configuration is about 11 kcal mol⁻¹ lower than that of the C_3v configuration. Therefore, we can expect that the employed BLYP functional and the TZP basis set with frozen-core approximation can result in reasonable structures and energies of all the configurations.

Table 2 Computed key geometric parameters and relative energies of the C₂ and C_3v isomers with different function

Method	Isomer	RLu–Lu (Å)	RLu–C (Å)	Relative energy (kcal mol^−1)
BP	C2	3.37	2.40	0.0
	C3v	3.49	2.44	11.5
PBE	C2	3.38	2.40	0.0
	C3v	3.50	2.44	10.6
PW91	C2	3.37	2.40	0.0
	C3v	3.49	2.44	11.3

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The key point of the controversy about the electronic structure assignment ((Lu₂)⁶⁺@C₇₆⁶⁻ by Umemoto et al. vs. (Lu₂)⁴⁺@C₇₆⁴⁻ by Yang et al.) is whether the two encapsulated Lu atoms form a single bond. To address this issue, we performed QTAIM analysis, which is a well-established method to analyze the topology of the electron density, and has been used for revealing and quantifying the bonding situation between the metal atoms and carbon cage in EMFs.^46–53 The result of the above obtained lowest-energy isomer of Lu₂@C₇₆ was shown in Fig. 3. Herein we focus on the encapsulated Lu atom pair. Our computations showed that there is one Lu–Lu bond critical point (BCP). The small values of electron density ρ_bcp (0.058 a.u.) and Laplacian ∇²ρ_bcp (0.005 a.u.) suggest weak covalent interaction between the two Lu atoms. The covalent nature is further confirmed by the negative total energy density (−0.021 a.u.) and the small ratio of the kinetic energy density G to ρ (<1). The covalent bonding between the encapsulated Lu pairs leads to the formal valence state of Lu₂⁴⁺@C₇₆⁴⁻, instead of Lu₂⁶⁺@C₇₆⁶⁻.

Fig. 3 Molecular graphs of the most stable Lu₂@C₇₆ (C and Lu atoms are black and white balls, respectively, the BCPs are the tiny spheres in red). Ring and cage CPs are omitted for clarity.

With the formal valence state of Lu₂⁴⁺@C₇₆⁴⁻, four electrons should be transferred from the encapsulated Lu pair to the outer C₇₆ cage. What factor is stabilizing the C₇₆(T_d) tetranion? Noting that aromaticity is playing an important role to stabilize spherical clusters,^54,55 we compared the relative energies and computed the NICS values at the structural centers of low-lying C₇₆⁴⁻ and C₇₆⁶⁻ isomers (Table 3). Our computed relative energies are in good agreement with those reported by Yang et al. Among the four structural isomers we considered, the hexanions of three isomers (with D₂, C_2v and C_s symmetry) have more negative NICS values, thus higher aromaticity, than the corresponding tetranions. The only exception is the T_d isomer, its C₇₆⁴⁻ has more negative NICS value than C₇₆⁶⁻, implying the stronger aromaticity of C₇₆⁴⁻(T_d), which should be partially responsible for its higher stability (Table 3), and echoes gracefully the formal valence state of Lu₂⁴⁺@C₇₆⁴⁻ instead of Lu₂⁶⁺@C₇₆⁶⁻.

Table 3 Computed relative energies (at B3LYP/6-31G*) and NICS values (at GIAO-B3LYP/6-31G*//B3LYP/6-31G*) of C₇₆⁴⁻ and C₇₆⁶⁻isomers

C76 spiral ID	PG	PA	C76^4−		C76^6−
			ΔE (kcal mol^−1)	NICS (ppm)	ΔE (kcal mol^−1)	NICS (ppm)
19151	Td	0	0.00	−7.93	6.83	−5.79
19150	D2	0	37.87	−12.91	31.58	−18.21
19138	C2v	1	17.12	−4.96	4.55	−14.00
17490	Cs	2	20.74	−11.36	0.00	−20.10

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The partial charges on the two Lu atoms obtained from our natural population analysis are both +1.5 |e|. The large charge transfer phenomena were also found in other EMFs.^56,57 For example, Popov and Dunsch's comprehensive theoretical analyses found that the metal atom charges in M₃N@C_2n (M = Sc, Y, La) are in the range of +1.6 to +1.9 |e|, the electronic structure of (M₃N)⁶⁺@C_2n⁶⁻ was assigned.⁵⁷

The transition state and motion trajectory

In Umemoto et al.'s study,²² they speculated that Lu atoms in the C₇₆ cage move rapidly in a rhombic dodecahedron trajectory to maintain the molecule's T_d-symmetry. However, Yang et al.²³ suggested that a tetrahedron trajectory. What is the true trajectory?

In order to compute the trajectory of Lu atoms, we located the molecular transition state between two optimized C₂ configurations. The obtained transition state has a C_s symmetry, the two Lu atoms are separated by 3.75 Å, and the smallest distance between Lu and the carbon cage is 2.42 Å. Fig. 4a and b are the top and side views of the transition state structure, respectively. The nature of the transition state was characterized by the only imaginary frequency of 54.8i cm⁻¹ in the DMol³ computations. The activation barrier is only 3.2 kcal mol⁻¹, which indicates that the two Lu atoms can hop from one C₂ point to another around the geometric center of carbon cage. On the basis of these data, we can get the partial trajectory that Lu atoms move inside the cage along the lowest energy path, as shown in Fig. 4c. We can further achieve the full motion trajectory by performing all the symmetry operations of T_d group to the partial trajectory based on the representative patch of the C₇₆(T_d). The obtained trajectory is a rhombic dodecahedron with O_h symmetry (Fig. 4d). Note that eight apexes of the trajectory are small planes consisting of Lu atoms (pointing to h1 or C1 in the representative patch), and they look like truncated corners. The motion trajectory of two Lu atoms is consistent with the STM image in Umemoto et al.'s study.

Fig. 4 Top (a) and side (b) views of the transition state structure; (c) the lowest energy path for the Lu atoms to hop between lowest-energy configurations, the blue balls represent the started C₂ isomer, the green balls represent C_s isomer in transition state, the yellow balls represent the terminated C₂ isomer, the arrow represent the motion direction; (d) the complete motion trajectory of Lu atoms in C₇₆ carbon cage.

Why does the motion trajectory have O_h symmetry when it is operated in the T_d group? There are two reasons:

(1). Not considering the outer C₇₆(T_d) carbon cage, the two Lu atoms are always moving around the center of symmetry i, which coincides with the geometric center O. By the symmetry operation:

S₄i = C₄, C₃i = S₆

All the symmetry operations of T_d group associated with the center inversion will generate the O_h symmetry.

(2). When taking the carbon cage into consideration, the Lu₂@C₇₆ molecule will show T_d symmetry due to the fast motion of Lu atoms inside the fullerene cage. As the T_d group is a subgroup of O_h group, the motion trajectory of Lu atoms can be reduced from O_h to T_d in the molecule. This trajectory is in good agreement with the experimental STM images.²²

¹³C NMR spectrum

Utilizing the representative patch of the C₇₆(T_d), the ¹³C NMR spectra of C₇₆(T_d) and Lu₂@C₇₆ can be qualitatively discussed. There are five kinds of carbon atoms (points C1, C2, C3, C4, C5) in the patch, thus, ¹³C NMR spectrum should have five peaks. Points C3 and C4 should give full intensity signals because of their local C₁ symmetry. In comparison, points C2 and C5 should present semi-intensity signals since they both have a mirror plane and local C_s symmetry, while with two mirror planes and a triple-axis, point C1 can only give 1/6 intensity signal due to its local C_3v symmetry. Based on the above analysis, the ¹³C NMR spectra of C₇₆(T_d) and Lu₂@C₇₆ should have five peaks: two full intensity signals, two semi-intensity signals and one 1/6 intensity signal.

To simulate the dynamic ¹³C NMR spectrum of Lu₂@C₇₆ quantitatively, we computed the average chemical shifts for all of the carbon atoms and identified their status. According to the representative patch of the C₇₆(T_d), carbon atoms are classified and their arithmetic average chemical shifts are calculated. The computed ¹³C chemical shifts and intensities are 126.55 ppm (C1 intensity 1), 134.36 ppm (C3, intensity 6), 135.51 ppm (C2, intensity 3), 140.72 ppm (C5, intensity 3), and 141.01 ppm (C4, intensity 6), which agree very well with the experimental data (Table 4) and those predicted by Yang et al.²³

Table 4 Comparison between the calculated and experimentally measured ¹³C NMR of Lu₂@C₇₆(T_d) chemical shifts (δ_cal and δ_exp) and intensities (I_cal and I_exp). The experimental results are from ref. 22

No.	δcal	δexp	Ical	Iexp	C
1	126.55	129.61	1	1.00	C1
2	134.36	134.51	6	6.67	C3
3	135.51	137.85	3	3.18	C2
4	140.72	140.65	3	3.23	C5
5	141.01	142.39	6	6.55	C4

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All the experimental data showed that Lu₂@C₇₆(T_d) has T_d symmetry,²² while our computations and those by Yang et al.²³ showed that its lowest-energy isomer has C₂ symmetry. How can we reconcile this discrepancy?

First we need to keep in mind that the experimentally measured UV-Vis-NIR, STM and ¹³C NMR chemical shifts present only the average result of the motion of Lu atoms in C₇₆ cage. Actually, two stable sites for Lu atoms exist inside C₇₆(T_d) cage, which leads to the lowest-energy isomer with C₂ symmetry. The small activation barrier (3.2 kcal mol⁻¹) enables these two Lu atoms hop between the two energetically most preferred sites inside C₇₆(T_d). Thus, it is the dynamical motion behavior of Lu atoms along the rhombic dodecahedron trajectory (with O_h symmetry) that results in the overall T_d symmetry of Lu₂@C₇₆.

The above conclusion that the dynamical motion of metal atoms inside the carbon cage can make the EMF molecule to maintain the symmetry of the parent fullerene cage is rather general. Similar situations were widespread in the other EMFs, and herein we give some examples. In M₂@C₈₀ (I_h, M = La, Ce) and Ce₂@C₈₀ (D_5h), the motion of metal atoms makes the whole molecule to maintain I_h or D_5h symmetry in the ¹³C NMR measurements.^{17,19–21,58,59} In M@C₈₂ (C_2v, M = La, Y, Sc, Gd), the motion of metal atoms makes the whole molecule to maintain C_2v symmetry.^{13–16,60–62} In M@C₇₄ (D_3h, M = Ca, Y, La, Ba and Sr), the energy minimum has a C_2v symmetry, while the ¹³C NMR showed D_3h cage symmetry due to the motion of metal atoms.^63–68 So far, all the experimental and theoretical results obey this rule. Finally, we propose that the overall symmetries of EMFs are determined by the symmetry of carbon cage, and the motion trajectory of metal atoms will act in concert with the cage to keep the symmetry.

Conclusions

By means of the relativistic DFT method and based on the representative patch of C₇₆(T_d), we obtained the lowest-energy isomer of Lu₂@C₇₆ (with a C₂ symmetry). Our QTAIM analysis and the aromaticity evaluations showed that the formal electronic configuration of Lu₂⁴⁺@C₇₆⁴⁻ rather than the traditional Lu₂⁶⁺@C₇₆⁶⁻ can be inferred, which supports Yang et al.'s electronic structure assignment of Lu₂⁴⁺@C₇₆⁴⁻.²³ The good electron delocalization in C₇₆⁴⁻(T_d) partially contributes to the thermodynamic preference of Lu₂@C₇₆(T_d). We further located the transition state structure for the Lu motion between the lowest-energy isomers, and obtained the motion trajectory of Lu atoms inside the C₇₆ cage. The rather small transition barrier leads to the flexible motion of the encapsulated Lu atoms inside the C₇₆ cage, and the computed O_h symmetrical motion trajectory of the two Lu atoms is consistent with the STM image. We also discussed the formation of this motion trajectory by group theory in detail. Additionally, the ¹³C NMR spectrum of Lu₂@C₇₆ quantitatively computed by this trajectory also agrees well with the experimental data. Our detailed analysis of the experimental and theoretical results indicate that that the overall symmetry of EMFs probably depends on the symmetry of carbon cage due to the dynamic motion of the encapsulated metal atoms.

Acknowledgements

This work has been supported in China by the National Natural Science Foundation of China (Grant nos 21036006, 21137001, and 21373042) and the State Key Laboratory of fine chemicals (Panjin) project (Grant no. JH2014009), and in USA by National Science Foundation (Grant EPS-1010094) and Department of Defense (Grant W911NF-12-1-0083). The computational resources were provided by Shanghai Supercomputer Center.

Notes and references

1. T. Akasaka and S. Nagase, Endofullerenes: A New Family of Carbon Clusters[C], Springer, 2002.
2. H. Shinohara, Rep. Prog. Phys., 2000, 63, 843.
3. X. Lu, L. Echegoyen, A. Balch, S. Nagase and T. Akasaka, Endohedral Metallofullerenes, Fundamentals and Applications, CRC Press, 2014.
4. S. I. Kobayashi, S. Mori, S. Iida, H. Ando, T. Takenobu, Y. Taguchi, A. Fujiwara, A. Taninaka, H. Shinohara and Y. Iwasa, J. Am. Chem. Soc., 2003, 125, 8116.
5. T. Tsuchiya, R. Kumashiro, K. Tanigaki, Y. Matsunaga, M. O. Ishitsuka, T. Wakahara, Y. Maeda, Y. Takano, M. Aoyagi, T. Akasaka, M. T. H. Liu, T. Kato, K. Suenaga, J. S. Jeong, S. Iijima, F. Kimura, T. Kimura and S. Nagase, J. Am. Chem. Soc., 2007, 130, 450.
6. R. B. Ross, C. M. Cardona, D. M. Guldi, S. G. Sankaranarayanan, M. O. Reese, N. Kopidakis, J. Peet, B. Walker, G. C. Bazan, E. V. Keuren, B. C. Holloway and M. Drees, Nat. Mater., 2009, 8, 208.
7. A. A. Popov, S. Yang and L. Dunsch, Chem. Rev., 2013, 113, 5989.
8. X. Lu, T. Akasaka and S. Nagase, Acc. Chem. Res., 2013, 46, 1627.
9. P. Jin, C. Tang and Z. Chen, Coord. Chem. Rev., 2014, 270–271, 89.
10. L. Dunsch and S. Yang, Small, 2007, 3, 1298.
11. L. Dunsch and S. Yang, Electrochem. Soc. Interface, 2006, 15, 34.
12. J. Tang, G. Xing, Y. Zhao, L. Jing, H. Yuan, F. Zhao, X. Gao, H. Qian, R. Su, K. Ibrahim, W. Chu, L. Zhang and K. Tanigaki, J. Phys. Chem. B, 2007, 111, 11929.
13. M. S. Golden, T. Pichler and P. Rudolf, Struct. Bonding, 2004, 109, 201.
14. W. Andreoni and A. Curioni, Phys. Rev. Lett., 1996, 77, 834.
15. E. Nishibori, M. Takata, M. Sakata, H. Tanaka, M. Hasegawa and H. Shinohara, Chem. Phys. Lett., 2000, 330, 497.
16. P. Jin, C. Hao, S. Li, W. Mi, Z. Sun, J. Zhang and Q. Hou, J. Phys. Chem. A, 2007, 111, 167.
17. T. Akasaka, S. Nagase, K. W. M. Kobayashi, K. Yamamoto, H. Funasaka, M. Kako, T. Hoshino and T. Erata, Angew. Chem., Int. Ed. Engl., 1997, 36, 1643.
18. E. Nishibori, M. Takata, M. Sakata, A. Taninaka and H. Shinohara, Angew. Chem., Int. Ed., 2001, 40, 2998.
19. J. Zhang, C. Hao, S. Li, W. Mi and P. Jin, J. Phys. Chem. C, 2007, 111, 7862.
20. M. Yamada, T. Nakahodo, T. Wakahara, T. Tsuchiya, Y. Maeda, T. Akasaka, M. Kako, K. Yoza, E. Horn, N. Mizorogi, K. Kobayashi and S. Nagase, J. Am. Chem. Soc., 2005, 127, 14570.
21. M. Yamada, N. Mizorogi, T. Tsuchiya, T. Akasaka and S. Nagase, Chem.–Eur. J., 2009, 15, 9486.
22. H. Umemoto, K. Ohashi, T. Inoue, N. Fukui, T. Sugai and H. Shinohara, Chem. Commun., 2010, 46, 5653.
23. T. Yang, X. Zhao and E. Osawa, Chem.–Eur. J., 2011, 17, 10230.
24. E. J. Baerends, J. Autschbach, A. Bérces, C. B. P. M. Boerrigter, L. Cavallo, D. P. Chong, L. Deng, R. M. Dickson, D. E. Ellis, M. van Faassen, L. Fan, T. H. Fischer, C. F. Guerra, A. S. J. van Gisbergen, J. A. Groeneveld, O. V. Gritsenko, M. Grüning, F. E. Harris, P. van den Hoek, H. Jacobsen, L. Jensen, G. van Kessel, F. Kootstra, E. van Lenthe, D. A. McCormack, A. Michalak, V. P. Osinga, S. Patchkovskii, P. H. T. Philipsen, D. Post, C. C. Pye, W. Ravenek, P. Ros, P. R. T. Schipper, G. Schreckenbach, J. G. Snijders, M. Sola, M. Swart, D. Swerhone, G. te Velde, P. Vernooijs, L. Versluis, O. Visser, F. Wang, E. van Wezenbeek, G. Wiesenekker, S. K. Wolff, T. K. Woo, A. L. Yakovlev and T. Ziegler, ADF2008.01, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, 2008.
25. G. te Velde, F. M. Bickelhaupt, E. J. Baerends, C. F. Guerra, S. J. A. V. Gisbergen, J. G. Snijders and T. Zifgler, J. Comput. Chem., 2001, 22, 931.
26. C. F. Guerra, J. G. Snijders, G. te Velde and E. J. Baerends, Theor. Chem. Acc., 1998, 99, 391.
27. B. Delley, J. Chem. Phys., 1990, 92, 508.
28. E. V. Lenthe, R. V. Leeuwen, E. J. Baerends and J. G. Snijders, Int. J. Quantum Chem., 1996, 57, 281.
29. E. V. Lenthe, J. G. Snijders and E. J. Baerends, J. Chem. Phys., 1996, 105, 6505.
30. E. V. Lenthe, E. J. Baerends and J. G. Snijders, J. Chem. Phys., 1994, 101, 9783.
31. S. H. Vosko, L. Wilk and M. Nusrir, Can. J. Phys., 1980, 58, 1200.
32. A. D. Becke, Phys. Rev. A: At., Mol., Opt. Phys., 1988, 38, 3098.
33. J. Perdew, Phys. Rev. B: Condens. Matter Mater. Phys., 1986, 33, 8822.
34. F. Biegler-König, AIM2000, version 1.0, University of Applied Sciences, Bielefeld, Germany, 2000.
35. R. F. W. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press, Oxford, U.K., 1990.
36. X. Cao and M. Dolg, J. Mol. Struct.: THEOCHEM, 2002, 581, 139.
37. D. A. Pantazis and F. Neese, J. Chem. Theory Comput., 2009, 5, 2229.
38. J. P. Perdew, Phys. Rev. B: Condens. Matter Mater. Phys., 1986, 38, 7406.
39. P. V. R. Schleyer, C. Maerker, A. Dransfeld, H. Jiao and N. J. R. V. E. Hommes, J. Am. Chem. Soc., 1996, 118, 6317.
40. Z. Chen, C. S. Wannere, C. Corminboeuf, R. Puchta and P. V. R. Schleyer, Chem. Rev., 2005, 105, 3842.
41. 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, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian 09, revision A. 01, Gaussian, Wallingford, CT, 2009.
42. K. Wolinski, J. F. Hinton and P. Pulay, J. Am. Chem. Soc., 1990, 112, 8251.
43. A. G. Avent, D. Dubois, A. Pénicaud and R. Taylor, J. Chem. Soc., Perkin Trans. 1, 1997, 2, 1907.
44. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
45. J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 6671.
46. A. L. Svitova, K. B. Ghiassi, C. Schlesier, K. Junghans, Y. Zhang, M. M. Olmstead, A. L. Balch, L. Dunsch and A. A. Popov, Nat. Commun., 2014, 5, 3568.
47. P. R. Varadwaj, I. Cukrowski and H. M. Marques, J. Phys. Chem. A, 2008, 112, 10657.
48. P. R. Varadwaj and H. M. Marques, Phys. Chem. Chem. Phys., 2010, 12, 2126.
49. B. Tan, R. Peng, H. Li, B. Wang, B. Jin, S. Chu and X. Long, J. Mol. Model., 2011, 17, 275.
50. A. A. Popov and L. Dunsch, Chem.–Eur. J., 2009, 15, 9707.
51. P. R. Varadwaj and H. M. Marques, Theor. Chem. Acc., 2010, 127, 711.
52. P. R. Varadwaj, A. Varadwaj, G. H. Peslherbe and H. M. Marques, J. Phys. Chem. A, 2011, 115, 13180.
53. Z. Badri, C. Foroutan-Nejad and P. Rashidi-Ranjbar, Comput. Theor. Chem., 2013, 1009, 103.
54. M. Bühl and A. Hirsch, Chem. Rev., 2001, 101, 1153.
55. Z. Chen and R. B. King, Chem. Rev., 2005, 105, 3613.
56. S. Osuna, M. Swart and M. Solà, Phys. Chem. Chem. Phys., 2011, 13, 3585.
57. A. A. Popov and L. Dunsch, Chem.–Eur. J., 2009, 15, 9707.
58. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science, 2004, 306, 666.
59. Z. Huang, X. Liu, K. Li, D. Li, Y. Luo, H. Li, W. Song, L. Chen and Q. Meng, Electrochem. Commun., 2007, 9, 596.
60. E. Nishibori, S. Narioka, M. Takata, M. Sakata, T. Inoue and H. Shinohara, ChemPhysChem, 2006, 7, 345.
61. L. Liu, B. Gao, W. Chu, D. Chen, T. Hu, C. Wang, L. Dunsch, A. Marcelli, Y. Luo and Z. Wu, Chem. Commun., 2008, 474.
62. S. Guha and K. Nakamoto, Coord. Chem. Rev., 2005, 249, 1111.
63. J. Xu, T. Tsuchiya, C. Hao, Z. Shi, T. Wakahara, W. Mi, Z. Gu and T. Akasaka, Chem. Phys. Lett., 2006, 419, 44.
64. H. Oliver, H. Martin, G. Arthur, M. Michael and D. P. J. Martin, Z. Anorg. Allg. Chem., 2005, 631, 126.
65. S. Ren, D. Tian and C. Hao, J. Mol. Model., 2013, 19, 1591.
66. T. Kodama, R. Fujii, Y. Miyake, S. Suzuki, H. Nishikawa, I. Ikemoto, K. Kikuchi and Y. Achiba, Chem. Phys. Lett., 2004, 399, 94.
67. W. Zheng, S. Ren, D. Tian and C. Hao, J. Mol. Model., 2013, 19, 4521.
68. S. Ren, D. Tian and C. Hao, Comput. Theor. Chem., 2013, 1020, 57.

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