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22.1 Introduction

Shortly after the discovery of multi-walled carbon nanotubes , by Iijima, fullerene analogs with negative Gaussian curvature have been proposed (Scuseria 1992), where the pentagons in fullerenes are replaced with large non-hexagonal rings (e.g., heptagons) (Terrones and Terrones 2003). Highly symmetric nanotube junctions , as tetrahedral or octahedral units, can be assembled into networks resulting into infinite periodic minimal surfaces (Diudea and Nagy 2007). Such junctions could play an important role in the use of nanotubes in microelectronic devices (Terrones et al. 2002a, b, 2003).

There are software programs (Brinkmann et al. 2010; Schwerdtfeger et al. 2013) that can be used for the generation of fullerene cages. However, since a nanotube junction corresponds to a spanned fullerene, a systematic design of multiterminal junctions , with a given number of terminals and molecular symmetry , is required for the appropriate transforming of fullerenes.

Tetrahedral fullerenes with an open-shell electronic configuration should be very reactive; therefore, the ground state will have a lowered symmetry because of the first-order Jahn–Teller distortion (Jahn and Teller 1937). However, the topological symmetry could be achieved in the case of ions, heteroatom-doped analogs, or adducts. For example, the tetrahedral isomer of fullerene C40 has only two electrons in the triply degenerated frontier orbitals. The tetra-anion C40 4−, the C36N4 nitrogen heteroanalog, and the C40H4 hydrogen adduct have a tetrahedral T d configuration. It was found that the two electron-rich species are also highly aromatic, with nucleus-independent chemical shift (NICS) (Schleyer et al. 1996) values in the cage center −15.7 and −11.8 for C40 4− and C36N4, respectively (Chen et al. 2001b).

Tetrahedral fullerene dication was also found to be highly stable (Diaz-Tendero et al. 2003, 2005). C52 2+ (T) has six adjacent pentagons and possesses lower energy than the isomers with only five adjacent pentagons, which reveals that the pentagon adjacency penalty rule (PAPR) (Campbell et al. 1996) does not necessarily apply to the charged fullerene cages. It is highly aromatic, as suggested by the −46.2 ppm negative NICS value (GIAO-6-31G*//B3LYP/6-31G*), and also very stable, as a consequence of the closed-shell electronic structure and nearly perfect sphericity.

The smallest tetrahedral fullerene has 28 carbon atoms and follows the 2(N + 1)2 rule for spherical aromaticity (Hirsch et al. 2000). The ground state has an open-shell configuration with four singly occupied frontier orbitals; therefore, it is considered to be tetravalent with four dangling bonds localized on the triplet-pentagon fusion apices and prefers to encapsulate metal atom inside to form endohedral metallofullerenes M@C28, as was observed in the gas-phase experiments (Guo et al. 1992). Upon reduction, the valence shell is completely filled, and the C28 4− anion is highly aromatic, as predicted by −35.5 ppm (GIAO–SCF/6–31G*//B3LYP/6–31G*) NICS value at the cage center (Chen et al. 2001a). The heteroatom-substituted C24N4 also has a high delocalization of the electrons as indicated by the −27.8 ppm NICS at the cage center (Chen et al. 2001a).

By means of group theory, the formulas for the number of IR and Raman active modes, for all of the tetrahedral fullerenes, have been obtained (Cheng et al. 1999).

22.2 Tetrahedral Carbon Cages

According to the distribution of carbon atoms, the tetrahedral fullerenes can be classified as follows: (a) cages with no atoms along the symmetry axis, (b) fullerenes with four atoms lying on the threefold axis, and (c) cages with each of the threefold axes crossing two carbon atoms.

According to the number of carbon atoms (n), the tetrahedral fullerenes have been divided into three classes. It was found that the electronic structure, evaluated by means of Hückel theory , varies according to (1) n = 12 N in closed-shell fullerenes, (2) n = 12 N + 4 in open shell, and (3) n = 12 N + 8 in open-shell or pseudo-closed-shell electronic configuration, where N represents a nonnegative integer determined by group properties (Tang and Huang 1996).

The 12 fivefold rings are divided, by the molecular symmetry , into four equivalent fragments, each lying along one of the four threefold symmetry axis. The four structural patterns that have been identified are shown in Fig. 22.1. Pattern A is a directly fused pentagon triple, and, as a consequence, these fullerenes do not obey the isolated pentagon rule (IPR) (Manolopoulos and Fowler 1992). With the increase of the number of atoms, the geometry of such a structure will become polyhedral, resembling a tetrahedron , as the giant icosahedral fullerenes resemble an icosahedron (Noël et al. 2014). Fragment B, commonly known as the sumanene, has a hexagonal ring core, surrounded by alternating pentagonal and hexagonal faces (with one exception, C28–T d). All fullerenes with B structural motif on at least 60 carbon atoms obey the IPR, and some of them are leapfrog fullerenes (Dias 1993). The other two classes have a carbon atom core and have isolated pentagons within the fragment.

Fig. 22.1
figure 1

Classification of the structural fragments decorating the tetrahedral carbon cages lying along the threefold symmetry axis. Some of the patterns have a common D fragment

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Notice that tetrahedral fullerenes must have two of these fragments, and accordingly they belong to two classes, e.g., in the case of the C40–T d fullerene, the C 3 axis crosses both an A and a B motif.

In the systematic built of all tetrahedral armchair [3,3] nanotube junctions by the transformation (spanning) of tetrahedral fullerene cages , it seems that a given junction can be derived from more than one cluster. The structural relationship between the different molecule series is listed in Table 22.1. Some of the fullerenes have icosahedral instead of tetrahedral symmetry (e.g., C20, C60, and C80); however, they arise by applying the structural transformation used in the other cases. Empty cells in the table represent non-fullerene polyhedrons.

Table 22.1 Classification of the tetrahedral fullerenes with respect to the structural fragment and the structural relationship with the corresponding tetrahedral armchair nanotube junction
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Geometry optimization followed by the calculation of harmonic vibrational frequencies was performed at the density functional level of theory (DFT ) using the hybrid B3LYP functional and the standard polarized double-zeta 6-31G(d,p) basis set. Single point energy calculation results are summarized in Fig. 22.2, where the total energy per atom and the HOMO–LUMO energy gap are plotted as a function of the number of carbon atoms. It can be observed that within series A, the charged fullerenes have very high energy gap values. The total energy per atom decreases with the cluster size. Within series B, the neutral structures show the highest stability according to the gap values, as these fullerenes have leapfrog structures.

Fig. 22.2
figure 2

Plot of the variation of the total energy per atom (E tot/N in a.u.) and the HOMO–LUMO energy gap (E gap in a.u.) as a function of the cluster size for the set of fullerenes in class A (top) and class B (bottom)

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22.2.1 C57 Multi-cage

The assembly of four C20 (I h) fullerene cages results in a tetrahedral multi-cage with 57 carbon atoms having exclusively five-membered rings (Diudea and Nagy 2012, 2013). Each cage shares the core vertex, such as C57 can be considered as a tetrahedral sp 3 carbon hyper-atom, where the cages correspond to the hybrid orbitals. The framework of the structure consists of a centrohexaquinane core (shaded rings in Fig. 22.3 left), with all tetravalent atoms, enclosed by four acepentalene (triple fused pentagon) fragments. The unit can assemble into hyperstructures, including the classical network of the diamond and lonsdaleite. Because of the high percentage of pentagons, these structures were generically called diamond D5 (Diudea et al. 2011, 2012).

Fig. 22.3
figure 3

Fragments of D5 network: the C57 (T d) multi-cage (left) and the corresponding hyper-adamantane C158 structure (right) (B3LYP/6-31G(d) optimized geometries)

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C57 has an open-shell electronic configuration, and the neutral structure lowers the symmetry to D 2d. The reduced form achieves maximal symmetry, and all fivefold rings belonging to the acepentalene fragment are equivalent. The electron-rich multi-cage has aromatic pentagons with NICS values comparable to the ones predicted for the acepentalene dianion C10H6 2−, −9.34 and −9.53 ppm, respectively. Strain relief by partial or total hydrogenation of the sp 2 carbon atoms was theoretically evaluated (Diudea and Nagy 2012).

The core of the adamantane hyperstructure C158 (Fig. 22.3 right) is the tetrahedral C28 (T d) carbon cage . In a larger network , the hexagons are shared between two C28 fullerenes.

22.3 Four-Terminal Nanotube Junctions

Multiterminal carbon nanotube junctions require, from a topological point of view, the insertion of structural defects in the hexagonal network of single-walled nanotubes . These defects are rings larger than hexagons which enables the assembly of individual nanotubes into a periodic network while preserving the sp 2 hybridization of the carbon atoms. In the following, tetrahedral zigzag and armchair nanotube junctions derived from fullerenes are presented, containing only heptagonal rings as defects.

22.3.1 Zigzag [3,0] Tetrahedral Junctions

The structure and stability of two tetrahedral three-terminal zigzag junctions consisting only of sp 2 covalent carbon atoms, Tj[3,0]40 and Tj[3,0]52, were investigated (Nagy et al. 2013) at the Hartree–Fock level of theory using the 6–31G(d,p) basis set as implemented in the Gaussian 09 software (Frisch et al. 2009). The structures were proven to be stable and possible to self-assemble in building diamond -like networks and hyper-dodecahedral architectures. Each opening of the junction fits a [3,0] zigzag single-walled carbon nanotube , found to be narrow-gap semiconductors (Borštnik et al. 2005). The junctions have been derived from the corresponding tetrahedral non-IPR fullerenes C44 (T) and C56 (T d) by the removal of the core carbon atom from each of the triple directly fused pentagons. Both tetrapodal junctions incorporate only hexagonal rings; however, in a network, three octagonal rings are necessary at each connection to give a periodic surface with negative curvature (Romo-Herrera et al. 2007). The optimized geometries of the junctions are shown in Fig. 22.4.

Fig. 22.4
figure 4

HF/6-31G(d,p) optimized geometries of tetrahedral zigzag nanotube junctions: Tj[3,0]40–T (left) and Tj[3,0]52–T d (right)

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The fullerenes C44 (T) and C56 (T d) both have, according to the simple Hückel theory , pseudo-closed electron configuration, while the corresponding junctions are pseudo-open. As a consequence, the highest topological symmetry can be achieved by addition of some negative charges. The hexagonal framework of both junctions consists of four phenalene fragments. These motifs share one bond with each neighbor, in the case of the Tj[3,0]40 junction, while in the case of Tj[3,0]52, the fragments are disjointly separated by one bond. The 13-carbon phenalene cation and anion motif has a diatropic perimeter ring current, as in case of coronene (Balaban et al. 2010), predicted by the ipsocentric model (Cyrański et al. 2007).

Computational data, obtained using HF/6-31G(d,p) method, are listed in Table 22.2 (Nagy et al. 2013). The strain energy, evaluated according to POAV theory (Haddon 1990, 2001) is also included. It can be observed that the closed fullerenes show a higher strain in comparison to the opened junctions. The higher symmetry of the charged Tj[3,0]40 4− results in a higher strain when compared to the neutral lower symmetry structure.

Table 22.2 Energetic data for fullerenes and the corresponding tetrahedral junctions, at HF/6-31G(d,p) level of theory
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As a consequence of the difference in the symmetry , the two tetrahedral junctions enable the building of two different networks by the assembly of the necessary number of units. In the case of the Tj[3,0]40 dimer, the openings (terminals) are oriented in a staggered conformation, which leads to the assembly of a hyper-hexagon (chair conformation). Furthermore, a hyper-adamantane and hyper-diamantane can be obtained, as shown in Fig. 22.5. In contrast to this, from the Tj[3,0]52 junction, a hyper-dodecahedron can be derived using 20 units, as a result of the occurrence of eclipsed terminals in the dimer (Fig. 22.6).

Fig. 22.5
figure 5

The hyper-diamond substructures designed by Tj[3,0]40 unit: hyper-adamantane Tj[3,0]40_ada (left) and hyper-diamantane Tj[3,0]40_dia (right)

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Fig. 22.6
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The junction Tj[3,0]52 enables the design of a hyper-dodecahedral architecture by joining 20 units (left). On the right side, a directly fused hyper-pentagon triple can be seen

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Details on the stability of some small structures built from the tetrapodal units can be found in Table 22.3, obtained at the HF/3–21G* level of theory (Nagy et al. 2013). Notice the overall structural relaxation increment observed in the strain energy with the increase in the number of joined units. According to HOMO–LUMO gap values, both hyper-ring structures have high kinetic stability.

Table 22.3 Computational data (HF/3–21G*) for smaller structures obtained by the assembly of tetrahedral nanotube junctions
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In open multi-tori built up from u tetrapodal junction units, the genus of structure is calculated as g = u + 1, irrespective of the unit tessellation (Diudea and Szefler 2012). In closed unit multi-tori, the genus can be calculated by formula \( g=u-{\displaystyle {\sum}_c\left({u}_c\times c/2\right)+1} \) where c represents the number of closures per unit, while u c is the number of units with c closures. Table 22.4 lists examples representing the proof of the above theorem (in agreement with the Euler formula) in both open and closed structures built up by the assembly of tetrapodal nanotube junctions (Nagy et al. 2013).

Table 22.4 Calculation of the genus in opened and closed adamantane- and diamantane-like hyperstructures (multi-tori ) by the Euler polyhedral formula and the number of repeat units
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22.3.2 Armchair [3,3] Tetrahedral Junctions

Armchair nanotube junctions were systematically designed starting from the corresponding tetrahedral fullerenes; the structural relationship is listed in Table 22.1. The structural transformation involves removal and/or addition of atoms, also modification of the connectivity . The first nine elements of the [3,3] tetrahedral nanotube junctions are shown in Fig. 22.7, where the position of the heptagonal rings is marked in blue (B3LYP/6–31G(d,p) optimized geometries). According to the Euler formula, it is necessary to introduce 12 heptagons to obtain a genus 2 structure.

Fig. 22.7
figure 7

DFT B3LYP/6–31G(d,p) optimized geometries of the armchair [3,3] carbon nanotube junctions . Position of the heptagonal rings is highlighted in blue

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The orientation of the openings enables the construction of dimers in staggered conformation, resulting a hyper-diamond network . The hydrogenated units were studied using DFT method; the single point results are plotted in Fig 22.8. All structures were geometry optimized in the highest symmetry configuration; therefore, negative charges were included as necessary in the case of open-shell structures. Notice that the electronic shell of the junctions corresponds to that of the class B tetrahedral fullerenes.

Fig. 22.8
figure 8

Plot of the variation of the total energy per atom (E tot/N in a.u.) and the HOMO–LUMO energy gap (E gap in a.u.) as a function of the tetrahedral junction size

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The smallest junction includes only heptagonal rings and can be derived from fullerene C28 by replacing the pentagons with sevenfold rings. The junction having the hexagonal framework of C60 fullerene is Tj[3,3]84 (T), which is the smallest leapfrog structure with isolated heptagonal rings.

In Fig. 22.8, one can observe that the stability ordering of the junctions is almost identical to the one observed in the case of class A fullerenes. The neutral tetrahedral junction Tj[3,3]84 (T) showed the highest gap value indicating that the cluster has a high molecular stability.

22.4 Conclusions

Tetrahedral fullerene cages have been classified according to the structural pattern found along the threefold symmetry axis and by the structural relationship with the corresponding nanotube junction. Armchair nanotube junctions having only heptagons and hexagons in the framework were built in a systematic way by spanning of the fullerenes. The supplied DFT results give an insight into the stability of tetrahedral nanoclusters. Diamond -like and dodecahedral hyperstructures were described from a topological point of view.