Introduction

Since the fullerene structure of C60 was postulated [1], much attention has been directed towards the novel nanostructures of carbon [2–5]. The highly symmetrical carbon fullerenes or clusters [6–15] with a hollow cage, including structures with I h symmetry, are of both fundamental and practical interest because of their larger surface area (compared to their compact-structure counterparts) as well as their potential applications as cages to accommodate other atoms or molecules, or as structural motifs to build highly stable core–shell nanoclusters [16]. For example, Romer et al. [17] presented ab initio calculations on endohedrally doped C28 solids, and found the structure of Zr@C28 will have a superconducting temperature T c lower than that found in K3C60 [18]. Yang et al. [16] investigated the atomic geometries, energetic stabilities and electronic properties of \( {\text{TM}}@{\text{Zr}}_{12}^{\text{z}} \) (TM = Y-Cd, Z = 0, ± 1) clusters, and found these clusters could be stable in I h symmetry. Owing to the hollow-cage structure characteristic, the capability of the fullerenes for trapping a variety of individual atoms or small molecules inside has caught the attention of chemists, physicists and material scientists since the discovery and macroscopic scale synthesis of C60 fullerene.

The C20 cage, consisting solely of pentagons, is the smallest unconventional fullerene which breaks the “isolated pentagon rule”. This fullerene is expected to have larger vibronic coupling than C60. Therefore, the solid form of this fullerene is a good candidate for a superconductor [19]. Among the many possible isomers of C20, it is not obvious that this highly symmetrical I h fullerene cage should be stable, and theoretical studies have reached different conclusions and argued for the resulting lower symmetry to be C 2 [20], D 5d [21], C 2h [22], C i [23] and D 3d [19, 24–26]. Unlike C60, dodecahedral C20, in particular the one with I h symmetry, is not spontaneously formed in condensation or by cluster annealing processes [27]. Despite this, the preparation of the C20 fullerene, by brominating dodecahedrane in the gas phase, has been tentatively identified and characterized [28]. The characterization of the fullerene from the above experiment was based on anion photoelectron spectroscopy. However, assignment of photoelectron spectra is sometimes controversial and the origin of this small spacing should be clarified so that the smallest fullerene can be clearly identified, since it was not clearly revealed by experimental study.

In the present paper, by placing interstitial atoms at the center of the C20 cage, we have tried to stabilize the highly perfect I h symmetry C20 fullerene cage. The alkali metals have been tested, resulting in the present investigations of the structural characteristics of M@C20 (M = Li, Na, K, Rb and Cs) endohedral metal clusters. We have performed structural optimization and frequency analysis of the M@C20 structures to confirm their energetic stability. The geometrical and electronic properties are described in terms of their binding energies (BEs), total magnetic moments (TMMs), highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) as well as the HOMO-LUMO gap (HLG). The charge distributions on atoms and the degeneracy and fractional occupation of the molecular orbitals are used to analyze the electronic properties of the M@C20 structures.

Computational Details

First-principles all-electron calculations of the total energies and optimized geometries were performed using density-functional theory (DFT) as implemented in the DMol3 software [29]. The exchange correlation energy was calculated using the generalized gradient approximation (GGA) [30] using the parametrization of Perdew and Wang [31] (PW91). Double numerical basis sets including polarization functions on all atoms (DNP) were used in the calculations. The DNP basis set corresponds to a double zeta quality basis set with a p-type polarization function added to hydrogen and d-type polarization functions added to heavier atoms. The DNP basis set is comparable to the 6-31G** Gaussian basis sets [32], with a greater accuracy for a similar basis set size [29]. The energy tolerance in the self-consistent field calculations was set to 10−6 Hartree. Optimized geometries were obtained using an energy convergence tolerance of 10−5 Hartree with the force on each atom less than 5.0 × 10−4 Hartree/Bohr. The frequency analysis has been performed to validate the energetic stability of all the clusters. The vibrational spectra are calculated in the harmonic approximation using the finite displacement technique to obtain the force constant matrix. The maximum allowed change in any Cartesian coordinate was set to 0.002 Å in order to obtain more accurate geometries. In order to achieve the SCF convergence when the gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital (HOMO-LUMO) is small, thermal smearing using finite-temperature Fermi function of 0.004 a.u. is used. The reliability of our method is tested by benchmark calculations on the C60 fullerene. Our current calculations demonstrate that the bond length of the 6:6 ring bonds (between two hexagons) and the bond length of the 6:5 bonds (between a hexagon and a pentagon) are 1.397 and 1.450 Å, respectively, close to the values 1.355 and 1.467 Å obtained by Liu et al. [33], and our HLG of 1.671 eV is quite close to the value 1.67 eV obtained by Diener and his co-worker [34].

Results and Discussions

Before studying the M@C20 endohedral fullerenes, we performed calculations for the C20 cage. As with reports in the literature [20–26], the present calculations show that the I h symmetry geometry of the pure C20 cage is not stable. However, the structure with D 3d symmetry, which is reported as the most stable structure [19, 24–26], is also obtained at the present computational level. The results for this structure will be used as a contrast in the following analysis. Considering the advantages of the I h symmetry structure, we try to stabilize this kind of C20 geometry by inserting other atoms at the center of the C20 cage keeping the symmetry unchanged. The simplest atom H is tested first. Unfortunately, the doped H@C20 cluster with I h symmetry is still unstable from the present calculation. Therefore other alkali metals are considered. By the way, we also test the doped clusters M@C20 with symmetry C i , D2h , D3d and D5d. The total energies of the optimization geometries are collected in Table 1. The total energies of some isomers are lower than the energies of their corresponding I h geometries, but they are not energy stable geometry because some imaginary vibrational frequencies occur. Only the C i structure of Cs@C20 shows no imaginary vibrational frequencies, while its energy is higher than that of I h structure. Hereinafter we only discuss M@C20 with I h symmetry.

Table 1 The total energies of the considered optimization geometries of X@C20 (in hartree)
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Geometries and BEs

The I h symmetry C20 structures with alkali metal atoms inserted at the center, i.e., M@C20 (M = Li, Na, K, Rb, Cs), are optimized at the PW91/DNP computational level. A frequency analysis for these shows that all the perfect I h symmetry structures are stable because no imaginary frequencies are found in the calculations. Because of the high symmetry, the geometries of the structures can be simply described by the distance between two nearest atoms on the surface (DS) of the cage, as shown in Table 2. However, the bond lengths of pure C20 with D 3d symmetry are different, so we use their average value in the following discussion.

Table 2 DSs, BEs and charge distributions of M@C20 (M = Li, Na, K, Rb, Cs) clusters with I h symmetry and pure C20 cage with D 3d symmetry
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From Table 2, we know that the DS gradually increases with increasing atomic number of the inserted atom, and all are larger than the average bond length 1.453 Å of the pure C20 cage. It implies that the interstitial alkali metal atoms can make the M@C20 clusters stabilize but enlarge their geometrical sizes, which results in the BEs of the doped clusters being obviously smaller than the 6.771 eV of the pure C20 cage and displaying a decreasing tendency. The binding energy can be calculated by the formula

$$ E_{b} = \left| {E_{\text{cluster}} - \sum\limits_{i}^{n} {E_{{{\text{atom}} - i}} } } \right| $$

where E b is the binding energy of the cluster. E cluster represents the total energy of the cluster, and E atom−i is the total energy of the i-th atom. The total energies of the cluster and atoms are calculated at the same theoretical level (PW91/DNP in the present work).

The variation of DS and BE with the doped alkali atoms are clearly shown in Fig. 1. These facts can be understood through analysis of the Mulliken atomic charge distributions of these clusters. As we know, there are 12 five-membered rings in the structure of C20 and each carbon atom supplies a single electron to the π-conjugation between the two neighbouring C atoms. The C20 cage possesses various cyclic paths through which π-electrons can delocalize. Figure 2a and b present the charge distributions from the Mulliken atomic charge analysis. Figure 2a shows that not all the carbon atoms in the pure C20 cage are neutral with the charges ranging from −0.014 to +0.021e. These unequal charges will strengthen the electronic attractive interaction between carbon atoms and result into shorter carbon–carbon bond lengths and bigger BEs.

Fig. 1
figure 1

a Ds of the M@C20 clusters and b BEs of M@C20 clusters

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Fig. 2
figure 2

a Pure C20 cage with D3d symmetry and its charge distribution and b Li@C20 cluster with I h symmetry and its charge distribution

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When the alkali metal atom is doped into the center of the cage, the charges redistribute on the atoms of the cluster. The results in Table 2 show that some charge is transferred from the interstitial alkali metal atom to the C20 cage and distributed equally between the carbons, so the interstitial alkali metal atom at the center of cage becomes a metallic cation. For example, the charge distribution of the stable Li@C20 cluster is found to be −0.018e on each C atom and +0.353e on the Li atom. For the sake of clarity, these are shown in Fig. 2b. Because of the equal negative charges on every carbon atom, a slight coulombic repulsion effect exists between the carbon atoms on the surface, which results in longer carbon–carbon bond lengths and smaller BEs. At the same time, a stronger symmetric attractive interaction occurs between the positive charge on the interstitial atom at the center and the negative charges distributed on each carbon atom. The two interactions make the I h symmetry M@C20 clusters have larger DSs and smaller BEs than the most stable pure C20, moreover they can be stable energetically.

HOMO-LUMO Energy Gaps and Magnetism

The electronic properties of the M@C20 clusters can also be investigated by examination of the HOMO and LUMO, HLG and TMM. In Table 3, we give the energy of the HOMOs, LUMOs, HLGs and TMMS of all the stable geometries, together with the occupation numbers, symmetries and degeneracies of the orbitals.

Table 3 HOMOs, LUMOs, HLGs and TMMs of the pure C20 cage and M@C20 clusters
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By comparison with C60 fullerene’s HLG of 1.67 eV [34], the current results show that the pure C20 cage with D 3d symmetry has an HLG value of 0.564 eV which is slightly smaller than 0.96 eV obtained by Sawtarie et al. [35] using a local density approximation (LDA) calculation. It implies that C20 fullerene could have better electrical transport characteristics than C60 fullerene, which is in accordance with that reported by Yoshiyuki et al. [36] that condensed phases of all-pentagon C20 cages can be possible superconductors. All the HLGs of the M@C20 clusters are smaller than that of the pure C20 cage with D 3d symmetry at the same computational level, which suggests that these doped clusters are more likely to be superconductors.

Now we consider the magnetic properties of these clusters. To obtain the TMM of the cluster, we calculate the spin up/down occupations of the molecular orbitals according to the Mulliken analysis. Then we obtain the magnetic moment of each atom by summing the spin occupations. Finally, the TMM of the clusters are obtained by summing all the spin occupations of the atoms. The TMMs are collected in Table 3. For the C20 fullerene, consisting solely of pentagons, the present calculation results show that all the spin up and down orbitals are in pairs and its TMM is 0.000 μB. Interestingly, the doped M@C20 clusters demonstrate different TMMs. We can see from the table that all the M@C20 clusters show stronger magnetism than that of the pure C20 cage. In particular, the TMM of Rb@C20 reaches 3.675 μB. It shows that the doped alkali metal atoms have strong and different effects on the magnetism of the C20 cage, which may produce many possible applications. To understand the different TMMs of the M@C20 clusters, we examine the occupation numbers of the HOMOs. We find that the orbitals of the I h symmetry M@C20 structures have high degeneracies because of the high symmetry of the clusters, which, in turn, is responsible for the different magnetism of the clusters. The degeneracies and occupation numbers of the HOMO and LUMO for the clusters are summarized in Table 3. In some cases, the degeneracy of the orbitals brings that one electron occupies equally two or more orbitals, which will raise the energies of the orbitals and result small HLG finally. The HOMO of the Cs@C20 is the 175-th molecular orbital of the cluster, which is a threefold degenerated molecular orbital (T 1u ) of I h point group. The degeneration raises the energy of the molecular orbital(HOMO) obviously, which makes HLG of Cs@C20 decrease to 0.047 eV and the energy gap between HOMO and HOMO-1 increase to 1.2 eV. However, for C20 cluster, two electrons occupy the two degeneracy orbitals of the HOMO, which makes up a pair of valence orbitals and gives a TMM of zero. The Li@C20 cluster has 123 electrons. However, its 121st–124th orbitals are of 4-fold degeneracy with G u symmetry, so the total number of occupied orbitals is 124 and the HOMO is not one orbital but 4 alpha-spin orbitals with fourfold degeneracy. Moreover, the alpha-spin and beta-spin orbitals are not all paired. The four unpaired alpha-spin orbitals result in the larger TMM of the Li@C20 cluster. This situation is also true for other M@C20 clusters. So, the strong magnetism of M@C20 clusters may be helpful for designing molecular devices or magnetic functional materials based on the C20 fullerene molecule. In brief, the various occupations of the molecular orbitals, in particular the different degenerated occupations, result in the colorful characteristics of the M@C20 clusters.

Conclusion

Pure C20 cages with D 3d symmetry and M@C20 (M = Li, Na, K, Rb, Cs) clusters with I h symmetry are investigated by DFT based on the GGA. The present calculations demonstrate that alkali metal atoms can stabilize the unstable C20 cage with I h symmetry by doping at the center of the cage. The stable effects can be understood through analyzing the Mulliken population of charges on the carbon and alkali metal atoms. Because of the fractional occupation numbers and the high degeneracy HOMOs, the HLG and magnetic moments of M@C20 clusters are clearly affected by the interstitial atoms. It implies that the high symmetry M@C20 clusters, which have rich and colorful molecular properties, can play an important role in designing novel molecular electronic devices.