Quantum-Mechanical Calculations on Molecular Substructures Involved in Nanosystems

In this review article, four ideas are discussed: (a) aromaticity of fullerenes patched with flowers of 6-and 8-membered rings, optimized at the HF and DFT levels of theory, in terms of HOMA and NICS criteria; (b) polybenzene networks, from construction to energetic and vibrational spectra computations; (c) quantum-mechanical calculations on the repeat units of various P-type crystal networks and (d) construction and stability evaluation, at DFTB level of theory, of some exotic allotropes of diamond D5, involved in hyper-graphenes. The overall conclusion was that several of the yet hypothetical molecular nanostructures herein described are serious candidates to the status of real molecules.


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silicates, etc., of many metals, have also found applications as nano-structured functional materials [4,5]. It is nowadays a common fact that nanotechnologies and nanomaterials have a great impact in the development of almost every industry [6][7][8][9][10]. The demands for new materials with adjustable properties have increased the interest for the study of possible nanomaterial precursors.
The polyaromatic hydrocarbons, PAHs, gained a new dimension by their recognition as precursors of graphene, and hence, of any of the abovementioned carbon allotropes. The progress in electronics and microscopy made possible new investigations and applications of very promising molecular structures. Quantum calculations would be useful in providing the theoretical background for new syntheses and applications.
This review article is organized as follows: after a very short introduction, the second section introduces to the design of new fullerenes patched with various circulenes, called here flowers, and their aromaticity vs. stability/reactivity is discussed in terms of HOMA and NICS parameters, based on Hartree-Foch and DFT calculations. Section 3 deals with polybenzenes, periodic nanostructures together with their repeating units, as finite molecules, optimized also at the Hartree-Foch and DFT levels of theory. In Section 4, some 3-periodic nanostructures built up by small units designed by opening spherical fullerenes are presented. In Section 5, some exotic molecules, involved in 2D-and 3D-nanosystems, are presented and their relative stability is evaluated at the DFTB level of theory. The review is ends with conclusions and references.

Circulene Patched Fullerenes
The specific properties of fullerene-based nanomaterials are mainly due to their π electron systems, so investigating the aromatic character of fullerenes and their precursors may offer a good insight into the nanomaterial properties. The aromatic character is a molecular property, conditioned by energy, electronic structure, magnetic response, geometric characteristics or chemical behavior [11][12][13]. Accordingly, various orderings are expected in sets of molecules with respect to different parameters, broadly called aromaticity criteria.
In the energetic criterion, a more aromatic character means a more stable structure [13]. Despite the fact resonance energy [14][15][16] plays an important role in stabilizing (at least) planar polyhex structures, the strain appearing in fullerenes, nanotubes, etc., will decide the overall stability (and reactivity) of such molecules.
The electronic criterion requires π-electron delocalization [11,12] (and bond length equalization). However, aromaticity is a local property, in the sense that small benzenic or naphthalenic units, rather than larger circuits, will manifest in chemical reactions. The π-electron distribution can be presented in terms of the numerical Kekulé valence structures [17][18][19][20][21][22][23][24] that, in contrast to geometrical Kekulé structures (e.g., the icosahedral C60 has 12,500 such structures), enable the construction of a single numerical structure to account for the superposition of the geometrical Kekulé structures, as in the Clar representation [25,26]. With regard to stability, a higher Kekulé structure count K is associated with a higher stability [11,27]. There are, however, twenty C60 isomers with K > 12,500, although they are less stable [28][29][30]. As mentioned above, the strain in the σ-frame could be a more important energetic factor, particularly in non-planar molecules like fullerenes, nanotubes, etc., where it may change the expected (by aromaticity) ordering. Thus, K alone seems not to be a reliable predictor of energetically favorable structures, and the conjugated circuits count has proven to be a more adequate description [11,[31][32][33].
The magnetic criterion describes the π-electron delocalization, with direct consequences on the magnetic properties, e.g., the diamagnetic susceptibility and NMR chemical shifts. These effects can be rationalized in terms of ring currents induced by the external field. Ring-current effects have long been recognized as important indicators of aromaticity. Depending on the number of π-electrons, diatropic or paratropic ring currents may occur. In fullerenes, enhanced aromaticity, as assessed by magnetic criteria, does not necessarily imply additional stabilization. The considerable strain of the σ-frame may dominate the stability and reactivity [11].
To conclude, aromaticity is a multi-dimensional phenomenon [42][43][44]. Fullerenes rather show an alkenic character [28], with additions being the most favored reactions. The electron deficiency of fullerenes results from the presence of the 12 pentagons (appearing as defects in the graphite sheet) needed to close the cage.
A circulene is a flower-like molecule, with a core and surrounding petals and general formula [n:(p1,p2)n/2], where n is the size of the core polygon and pi are the polygonal petals. For n < 6, the molecule has a bowl-shaped geometry whereas for n > 6 it is saddle-shaped [1,2,45]. The bowl-shaped circulenes are potentially useful in the direct synthesis of fullerenes [46,47] while the saddle-shaped ones would appear as patches in the foamy structures of spongy carbon [48,49]. The idea of increasing the stability of fullerenes tessellated by disjoint circulenes/flowers originates in the classical texts of Clar [25,26] who postulated disjoint benzenoid rings as a criterion for aromatic conjugation [50]. Figure 1 shows two types of fullerene covering, one with joined patches and the other one with disjoined patches. The aromatic character of various flowers, with the core being either a hexagon or an octagon, has been evaluated by means of geometric (HOMA index), energetic (heats of formation) and magnetic (NICS index [51] and exaltation of magnetic susceptibility) criteria.

Global Stability
In order to evaluate the stability of the considered polycyclic compounds, the HOMO-LUMO HL gap and total energy per C-atom were computed [52] (Table 1). HOMO-LUMO gap may be considered as an approximation to the chemical hardness and also an indicator of the molecular kinetic stability. Larger values of HOMO-LUMO gap are found for coronene and sumanene that suggest a higher stability for these two experimentally known molecules.

Aromaticity
The local aromaticity of these circulenes can be evaluated by calculating the NICS(0) and NICS(1) indices for every ring of the polycyclic hydrocarbons [52]. Results of these calculations are shown in Tables 2-6. For coronene, the NICS data show a pronounced aromatic character of the outer benzenic rings and lower aromatic or even non-aromatic character of the core hexagon. These data support the "radialene"-structure of coronene, as depicted in Figure 2 (left). The HOMA values also show an enhanced aromaticity on the outer rings [52].
It should be noted that coronene itself is not a totally resonant hydrocarbon [11,53] because every Kekulé structure leaves some carbon atoms outside the sextet rings. However, Clar [26] proposed that if the three sextets of coronene can migrate into the neighboring rings, an extra ring current would emerge. The sextet migration current can be taken as an argument in favor of the enhanced aromaticity of coronene (compared to some other polycyclic hydrocarbons, e.g., naphthalene and anthracene) [11].   Computations of the NICS(0) index for [6:(5,7)3] isocoronene ( Figure 2, middle) provide close values for the central 6-membered and the 5-membered rings of this polycyclic structure, the rather low negative values indicating a low aromatic character [52] (Table 3). The NICS(0) positive values of the 7-membered rings suggest a non-aromatic character. The NICS(1) index is often employed as an indicator of the π-electron delocalization; in the case of 6-and 5-membered rings of isocoronene, it provides "more negative" values. The enhanced values are attributed by Fowler et al. [54] to the electron flow through the outside perimeter of the rings. On the other hand, the HOMA values show a different trend compared to both of the NICS indices, suggesting a more pronounced aromatic character of the central benzenic ring (see also [55]).
The values of the NICS(0) and NICS(1) indices for sumanene ( Figure 2, right) correspond to an anti-aromatic character of the pentagons, a strong aromatic character of the outer benzene rings and a lower aromatic character of the core R [6] ring (Table 4). The HOMA data closely parallel their NICS counterparts [52]. Table 6. Aromaticity (HOMA and NICS indices) and strain (POAV, kcal/mol) of coronene and sumanene patches in the tetrahedrally spanned fullerenes (Figure 3) (optimised at HF/6-31G(d,p); B3LYP/6-31(d,p) levels of theory) [52].  The coronene and sumanene fragments can be inserted into 3D-structures such as the tetrahedrally spanned fullerenes depicted in Figure 3. These structures can be derived from the fullerene C84 and were named Cor_T_84 and Sum_T_84, respectively; they can also be considered as junctions of nanotubes [1]. Even there are many tessellation for the tetrahedral nanotube junctions, our option was for these two patches, as they also represent real molecules. Data for these structures (as hydrogen-ended ones) are compiled in Table 5, in comparison to those for all-carbon C60(Ih), the most used reference structure in nanoscience [56]. It seems the two tetrahedral structures show a pertinent stability, when compared to that of the reference fullerene, with Sum_T_84 being particularly stable [52]. However, there is no direct comparison with C60 since there are known the differences, at least in HL gap, between the all-carbon-and hydrogen-ended structures.

Structure
The NICS(0) values in Cor_T_84 (Figure 3, left and Table 6) are in good agreement with those in the free coronene molecule ( Table 2). The DFT data show (in general) the same trend, with an even increased negative values of NICS indices. The index NICS(−1) refers to the "inside" while NICS(+1) refers to the "outside" of spanned tetrahedral fullerenes. The NICS(−1) negative values are larger than those provided by NICS(+1), indicating a higher conjugation of π-electrons inside the structure.
The HOMA values calculated for Cor_T_84 exhibit the same trend as the NICS(−1) values, namely the highest aromaticity of the free hexagons R6,plane, followed by the R6,Core and finally the bound hexagons. The HOMA index allows calculation for both a patch and the whole molecule (Table 6). However, this geometric criterion must be completed with other criteria of aromaticity when an ordering of molecules is attempted [52].
In Sum_T_84 (Figure 3, right), all the NICS values exhibit the highest aromaticity of the outer R6 rings in comparison to the core hexagon (see also Table 4). The pentagons appear rather anti-aromatic by NICS(−1) values but still aromatic by NICS(+1) values, (with lower values in comparison to the core hexagon). In case of HF-data, the HOMA vaues follow the trend of NICS(0) and NICS(−1) values while in the DFT-optimized structure the trend of HOMA values is different from that of NICS data.
The extent of strain, evaluated by POAV1 theory [57], varies among the rings. It is the largest for the bound-hexagons in Cor_T_84 and for the core hexagon and pentagons in Sum_T_84, but these values are even lower than those for C60(Ih) (8.256 kcal/mol) because these structures are "opened fullerenes". The extent of strain for the patch and the whole molecule are irrelevant. Since the NICS and HOMA calculations indicated the presence of some anti-aromatic substructures, it was necessary to recalculate the basic 6-flowers: coronene, isocoronene and sumanene, both in singlet and multiplet states [52] (Table 7). Table 7. Total energy, Etot (in au), total energy per C-atom, Etot/C, and HL gap (in eV), (HF and DFT (B3LYP/6-31(d,p)) for the 6-flowers, in singlet and multiplet states (in Italics) [52]. The calculations have shown that no important variation in HL gap values appear between the alpha and beta orbitals of the triplet states (in italics) of coronene and isocoronene molecules, as the conjugacy of the pi-electron was not deeply affected.
The sumanene triradical should be non-planar. Planarization induces in-plane symmetry breaking; as a consequence, the sumanene gap value presented in Table 1 was overestimated. The differences in HOMO-LUMO gap of the alpha and beta orbitals, in the higher multiplicity state, clearly indicates a lower conjugasy (and a lower aromaticity) for the sumanene structure.

Circulenes with Octagonal Core
In this section, the aromatic character of circulenes with octagonal core and petals consisting of 5-, 7-rings is discussed [58]. The three 8-flowers herein discussed are shown in Figure 4. The values of HOMA and NICS(0) indices and the magnetic susceptibilities have been computed; in addition, two different isodesmic reactions (for each molecule in Figure 4) have been proposed for calculating the enthalpies of formation. Electronegativity [59], total hardness [59], the electrophilicity index [59] and Fukui functions [60] (for an electrophilic attack) have been computed based on DFT methods.

Enthalpy of Formation (Energetic Criterion)
The stability of a polycyclic hydrocarbon can be investigated on the basis of some computed thermodynamic values, particularly the enthalpy of formation. Comparison between the calculated heats of formation of the circulenes [8:(5,7)4], [8:68] and [8:(5,6)4] and the experimental available data, e.g., for coronene, may lead to valuable conclusions regarding the stability of the three mentioned flowers. In this regard, two isodesmic reaction schemes for each of these circulenes were proposed [58] (Figure 5); the average heat of formation was calculated, as shown in Table 8.

NICS(0) Index and Exaltation of Magnetic Susceptibility (Magnetic Criterion)
NICS(0) index (or variants like NICS(0)πzz) [77] is widely used as a local descriptor of aromaticity. NICS(0) values were calculated for the 8-membered central ring of the studied circulenes, as well as for the petal-rings (Table 9) [58]. Table 9. NICS(0) values for the 8-flowers in Figure 4 (optimized at the B3LYP/6-31G(d) level) [58]. For comparisons, data for the petal rings in circulenes with hexagonal core [52] are presented in round brackets. Irrespective of the core size, the trend of NICS(0) values appears to be the same [58]. The largest aromatic character (i.e., largest negative values) was found in 6-atom petals of circulenes [8:68] Table 10 show that the 6-atom petals of [8:(5,6)4] flower have the most pronounced local aromatic character. Alternating values for the 6-atom petals of the saddle-shaped circulene [8:68] was observed, as shown in case of NICS(0) values (see Table 9). For the 5-and 7-atom petals, negative values (i.e., anti-aromatic character) were reported [58]. Besides their use in evaluating the reactivity and regioselectivity of chemical reactions, reactivity descriptors like absolute hardness (η), electrophilicity (ω) and Fukui functions have also been applied to evaluate the aromatic character of molecules [58,78]. The absolute hardness (η) is calculated as half of the HOMO-LUMO gap; a harder molecule is associated with an increased stability, so molecules with larger η values are believed to be more stable, thus showing a possible aromatic character. Also, a lower electrophilicity ω value can be taken as a proof of aromaticity. Regarding the local reactivity descriptors, the Fukui functions computed for an electrophilic attack are good indicators of reactivity of each atom in the studied circulenes, thus a hierarchy of the most electrophilic sites could be established. The above descriptors of reactivity are defined as follows: Absolute hardness [59]: Data for the discussed flowers are given in Table 11; they are in good agreement with the geometric, magnetic and energetic criteria above used to evaluate the aromaticity in the three circulenes  The Fukui functions [60] (for an electrophilic attack) have been computed for each carbon atom on the contour of circulenes optimized at (B3LYP/6-311G(d,p) level of theory ( Figure 6). Corroborating the results of different criteria of aromaticity on the three circulenes led to the conclusion that, the most "aromatic" one is the 8-sumanene, [8:(5,6)4], in agreement with the fact that tetraoxa [8]circulenes represent real molecules [63]. Following this result, two different fullerenes, bearing 6-sumanene and 8-sumanene patches, have been designed (Figure 7). The values of NICS(0; −1 (inside the cage); +1 (out of the cage)) and HOMA indices, for the 5-, 6-, and 8-rings of the fullerenes in Figure 7 are listed in Table 12.  In case of C120, the NICS(0) values for the 8-sumanene patch show the same trend as in the planar 8-sumanene. The values NICS(−1), characterizing the inside cage electron density, show more aromatic character (i.e., larger negative values) in comparison to the outside cage describing NICS(+1), as expected. Comparing the 6-sumanene patch in the two cages in Figure 7, one can see a more aromatic character of petals vs the core, according to NICS(0) values. The HOMA values follow in general the trend of NICS values, excepting the 6-sumanene patch in C120, where the 6-ring core was found more aromatic (i.e., more positive value, 0.821) than the 6-ring petal (0.537).
In order to estimate the stability of the C52 and C120 cages, two isodesmic reactions have been proposed [58] (Figure 8). The values of Hf (Table 12, 4th column) are in good agreement with the ones of the single point computations (Etot/C); there are no significant differences between the computed stability of the two structures. However, the HOMO-LUMO gap value is in favor of C120 (6.251 for C120 vs. 5.317 for C52).

Polybenzenes
O'Keeffe et al. [79] proposed, about twenty years ago, two 3D networks of benzene: the first one, called 6.8 2 D (also polybenzene, Figure 9), is described to belong to the space group Pn3m and having the topology of diamond. The second structure (Figure 10) was called 6.8 2 P and it belongs to the space group Im3m, corresponding to the P-type-surface. These networks represent embeddings [80] of the hexagon-patch in the two surfaces of negative curvature, D and P, respectively.
These triple periodic minimal surfaces (as in the soap foam) can embed networks of covalently bonded sp 2 atoms, called periodic schwarzite [1,45] in the honor of H. A. Schwarz [81,82] who, in the nineteen century, studied the differential geometry of such surfaces.
The networks were constructed [84] either by identifying or joining the common faces in the corresponding repeating units, BTA_48 and BCZ_48, respectively (Figures 9 and 10, left). Face identification in case of the armchair-ended, tetrahedral unit BTA_48 is possible either by octagons R (8) or by dodecagons R (12). Identification by R(8) of the BTA_48 units, disposed at the center of the six faces of Cube, leads to the 6.8 2 fcc-net (Figure 9, right), with the topology of D6-diamond; the corresponding R(8)-dimer we call the "dia-dimer" BTA2dia_88 ( Figure 11, top, left). When R(12) are identified, the resulting oligomers are dendrimers ( Figure 11, bottom row) and the R(12)-dimer is named "dendritic dimer" BTA2den_84 ( Figure 11, top, right). Dendrimers, after the second generation, completely superimpose over the BTA48_fcc-net (Figure 11, middle and bottom rows).   Evaluation of the stability of polybenzenes was performed on finite hydrogen-ended structures (Tables 13 and 14). Data include the total energy Etot, total energy per carbon atom, Etot/C-atom, HL gap, strain energy according to Haddon's POAV theory and HOMA index for the benzene patch R [6]; the reference structure was taken the fullerene C60(Ih). The trend of energy values is similar in HF and DFT approaches. Since no interactions with solvents are of interest here, as DFT approaches can evaluate, for structures of a large number of atoms only HF calculations have been performed. Table 13. Polybenzenes: total energy Etot; total energy/C-atom Etot/C-atom and HOMO-LUMO HL gap (at Hartree-Fock HF/6-31G(d,p) level of theory), strain (by POAV theory) and HOMA index, with C60(Ih) as the reference structure [84].

Structure
No  Among the structures considered in Table 13, the most stable appears the armchair-ended unit BTA_48, with a tetrahedral embedding of benzene patch (Table 13, entry 1), followed by BTA2dia_88 (Table 13, entry 3). The third is the dendritic dimer BTA2dend_84 while the stability of some oligomers (i.e., dendrimers) of BTA_48 decreases monotonically with the increase of number of composing units (Table 13, entries 4 to 7) as suggested by Etot/C-atom and HL gap. The strain of these dendrimers decreases with the increase in the number of their carbon atoms. This is reflected in the values of HOMA: the benzene patch seems to be little distorted from the ideal planar geometry, with a maximum at the dendrimer with a complete generation, e.g., BTA5dend_192 (Table 13, entry 7).
The BCZ_48 structure (Table 13, entry 2) shows the highest value of HOMA, even the benzene patch is less planar in comparison to the same patch in BTA_48; it is the most strained structure among the all ones in Table 13. It seems that the C_C bond length is not the only parameter reflecting the pi-electron conjugation, as limited by HOMA index. Looking at the data in Table 13, entry 8, the reference fullerene C60(Ih) appears the least stable among all the considered structures; recall that it is all-carbon and data cannot be directly compared to those of hydrogen-ended molecules. However, polybenzenes have the total energy per carbon atom close to that of the reference fullerene. For BTA_48, and BCZ_48 the simulated vibrational spectra are given below (Figures 12 and 13) [84].
From Table 14, one can see that, the "armchair" A-structures are more stable than the "zig-zag" Z-structures, according to their total energy per carbon atom and HOMO-LUMO gap values [85,86]. The difference observed between the two series A/Z comes out from the size of the opening ring: 12 in case of A-series and 9 in case of Z-series, even the patch is always a hexagon. The planarity of benzene patch (more planar in case of A-series, than in case of Z-series) will influence both the energetics and vibrational spectra (Figures 18 and 19) of these structures [85].  IR spectra of BTA-48 and BTA2ecl_90, respectively ( Figure 18, left), show in the region 400-720 cm −1 the following differences in the absorbance bands: 425 cm −1 band splits into 417 and 443 cm −1 ; 693 cm −1 band splits into 624, 671 and 705 cm −1 [87]. This splitting can be interpreted to account for the dimer joining bonds [85]. In the Raman spectra of BTA-48 and BTA2ecl_90 (Figure 19, left), common features for a benzene-like structure can be identified and an additional Raman signal around 1840 cm −1 for the dimer BTA2ecl_90 as well. In the Z-series, the IR spectrum (Figure 18, right) shows two intense peaks at 438 and 538 cm −1 that can be attributed to the dimer bonds. Raman band around 1575 cm −1 (Figure 19, right) corresponds to the C-C stretching of the phenyl ring [88,89]. The presence of the vibration modes around 1345 and 1470 cm −1 indicate the formation of the dimer [90].

P-Type Surface Coverings
In the experimental conditions of fullerene synthesis, it is possible that some cages appear spanned, the "open"-faces next suitably joining to each other to eventually form a nanotube. We call such spanned fullerenes "nanotube junctions" [1]. According to their symmetry, we can distinguish tetrahedral, octahedral and icosahedral junctions.
Tetrahedral junctions are particularly interesting due to their similarity with the tetrahedral sp 3 hybridized carbon atom: the valences are now nanotubes while the atom is an opened cage embedded in a surface of genus 2. Recall, an embedding is a representation of a graph on a surface S such that no edge-crossings occur [80]. Genus is the number of handles to be attached to the sphere to make it homeomorphic to the surface on which a graph was embedded, or the number of connections of a given surface (the reader can find more information about structures of high genera, in [1,45]). As the single C-atom, a tetrapodal junction can be used to build various nanostructures such as diamondoids and multi-tori. Octahedral junctions (of genus g = 3) appear in zeolites, of which associated graphs are embedded in the P-type surface. Icosahedral junctions are also possible, as they appear in icosahedral multi-tori [91]. Zeolites [83] are natural or synthetic alumino-silicates with an open three-dimensional crystal structure. Zeolites are micro-porous solids used as "molecular sieves".

Sumanene Including Structures
Sumanene can be used as a primary real molecule in the synthesis of some structural units: Sum_T_A_108, Sum_T_84, Sum_CZ_192, Sum_CA_216, and Sum_S2LeX_168 (Figure 20), that can next compose more complex nanostructures, e.g. ordered schwarzites, embedded in the P-surface [92].
Hypothetical crystal carbon networks can be built up from the units listed in Figure 20, either by identifying two opposite open faces (Figure 21, left), or by joining the opposite atoms ( Figure 21, middle and right), by the aid of Nano Studio software [93], that also enables their embedding in the P-type surface [1,2]; these networks belong to the space group Pn3m. Stability of the H-ended structures bearing the sumanene patch ( Figure 20) was evaluated; Table 15 lists the energetic data, obtained after optimization at Hartree-Fock (HF) level of theory [92].  From Table 15, it is clear that such molecular structures show values of Etot/C comparable to that of C60(Ih) reference structure; Sum_T_84 and Sum_S2LeX_168 are the most simple and stable units, possible candidates for laboratory synthesis. Table 15. Total energy, Etot (in au), total energy per C-atom and HL Gap (in eV), (at HF/6-31G(d,p) level of theory) for H-ended sumanene-patched structures in Figure 20 [92].

Spanned Cages Patched by Hexagons Only
A covering by a single type polygon is called a Platonic tessellation [1]. The units in Figure 22 were designed [94,95] either by using symmetry in embedding the triple hexagon patches (Figure 22, top row) or by applying map operations Op2a(S2(M)); M = tetrahedron T, or cube C (Figure 22, bottom row).
Observe the twisted/chiral appearance of these last units, about 90 degree in case of C_3HextwZ_80. More about the map operations can be found in [96][97][98]. The unit T_3HexZ_52 provides an "eclipsed" dimer, which can self-arrange to a hyper-pentagon, the join of 12 such hyper-faces leading to a multi-torus T_3HexZ20_1040 (not seen) of icosahedral symmetry. In the opposite, T_3HextwZ_40 forms an "intercalated" dimer, next leading to a hyper-hexagon, which can arrange in a diamondoid network [95], as shown in Figure 23. The unit C_3HexZ_104, containing triple hexagon 3f6 patches, forms a 3-periodic lattice, embedded in the P-surface ( Figure 24). The C_3HexZ network is a new one, designed by TOPO Group Cluj, a 4-nodal net of the Pm-3m group. It has the point symbol for net: {6.8 2 }3{6 2 .8}6{6 3 }4 and vertex symbol [6.6.6] [6.6.6] [6.6.8] [6.8.8]. Stability evaluation was done on H-ended molecules, optimized at Hartree-Fock level of theory; data are listed in Table 16. One can see that, the HOMO-LUMO gap (calculated at HF level of theory) is the highest for the reference fullerene C60(Ih) ( Table 16, entry 5), however, Etot/C-atom, is favorable to the "twisted" junctions (Table 16, entries 3 and 4), even the strain of these structures, calculated by POAV theory, is higher than for the non-twisted ones. The strain is lower for the octahedral junctions (Table 16, entries 2 and 4), as expected for structures with larger "open" faces. The HOMA index values follow the trend of strain data [94,95]. Similarly, the Kekulé structure count [11] is in favor of octahedral junctions. Figure 24. C_3HexZ_104, g = 3 (left), forms a P-type crystal network C_3HexZ_(3,3,3)_2808 (middle); the same network, shown in the corner view (right). Table 16. Triple hexagon-patched (H-ended) structures: total energy per C-atom, Etot/C, and HL gap (at (HF/6-31G**); strain energy per C-atom, (by POAV1); HOMA aromaticity index and Kekulé structure count [94,95]. Resuming, the junctions patched by triple hexagons show several stability parameters close to those of the reference C60(Ih) fullerene. In supporting the idea that various nanotube junctions could appear in real experiments, we simulated the vibrational spectra of these junctions (Figures 25 and 26). These spectra show clear differences between the two different embeddings (in tetrahedra and cubes, respectively) and also between twisted and non-twisted ones [94,95].

Nanotube Junctions Patched by Heptagons Only
The units obtained by applying the septupling Sk, k = 1,2 map operations on the cube C, can form translatable crystal networks, as illustrated in Figure 27. This is the already known kgn network, a 3-nodal one belonging to the group P432, with the point symbol for the net (7 3 ) and the vertex symbol [7.7.7] [7.7.7] [7.7.7]. It is related to the well-known Klein graph [1]. One can see the large hollows represent C_3HepA_104 (Figure 27, left, designed by Op2a(S2(C)) while the small hollows come from C_3HepZ_80 ( Figure 27, middle, designed by Op(S1(C)) [94].

RAMAN_C_3HextwZ_80
A stability test was done on H-ended molecules, optimized both at the HF and DFT levels of theory; data are listed in Table 17. From these data, one can see that the highest value, among the considered structures, for the total energy per C-atom was provided by the Platonic all-pentagon C20 smallest fullerene (Table 17, entries 1, 5); this is probably due to its huge strain, higher 3.3 times than that of C60(Ih) (Table 17, entries 4, 8). The pyramidalization of sp 2 C-atoms, as evaluated by the Haddon's POAV theory [57], is related to the strain energy appearing in the graphite sheet when it is forced to embed in the sphere (i.e., closed fullerenes) or in other surfaces (the case of open fullerenes, herein studied). An increased strain value suggests an increased percent of sp 3 -hybridized C-atom, reflected in the C-C bond length (see Table 17, the last two columns). The bond-length values suggest an extent of alternant double/single C=C/C-C bonds; none of the studied structures is significantly aromatic, in agreement with their HOMA index of aromaticity, that shows values less than 0.5 (1 being the reference benzene molecule). Since the pyramidalization angles can be calculated either on closed or open (end-hydrogenated) fullerenes, the strain energy data in closed/open structures can be directly compared [94]. Table 17. Triple heptagon-patched (H-ended) structures: energies (total energy per carbon atom Etot/C, HLGap, (at HF/6-31G** and B3LYP/6-311+G**, respectively); POAV strain energy per C-atom); HOMA index of aromaticity, Kekulé structure count, extreme C-C bond length, and averaged bond length, in Ang [94]. The lowest strained structures in Table 17 are the Platonic all-heptagon open units; this is due to the patch 3Hep = 3f7, either as the free molecule C16_3f7 (strain (HF): 0.044; strain(DFT): 0.089 kcal/mol) or included in these open fullerenes (Table 17, entries 2,3; 6,7). The strain value, in these two units, is two orders of magnitude lower than that in C60(Ih). The 3f7 patch behaves quite the same, irrespective of embedding: tetrahedron, T_3HepA_52 (g = 2, entries 2, 6) or cube, C_3HepA_104 (g = 3, entries 3, 7). The averaged C-C bond length values (Table 17, last two columns) show the lowest value for these all heptagon open fullerenes (1.406 and 1.408 Ang, respectively) supporting their lowest strain values. They also show the lowest values of Etot/C-atom, predicting a good stability of these yet hypothetical molecular structures.
HOMA geometric index values collected in Table 17 are irrelevant, suggesting a rather anti-aromatic character for these relaxed structures. Similarly, the values of Kekulé structure count, related to the conjugation of pi-electrons, suggest this phenomenon is less important in the studied structures, while the strain originating in the sigma bond skeleton is a dominant feature.
The simulated IR and Raman spectra (Figures 28 and 29) for these triple heptagon-patched open structures show all vibrations with no imaginary values, proving the optimized structures represent global minima [94]. Data collected in Table 18 show that the 3f7-patch has its fingerprint in IR spectrum; the peaks in bold-italic represent the most intense vibrations, useful as a marker, for an eventual experimental (quick) identification of these structures.   *: normal font = medium peak; bold = intense peak; bold&italic = marker peak.
Substructures of D5 are related to the classical D6 diamond [3]. An adamantane-like structure D5_ada can form two diamantane-like D5_dia forms ( Figure 29, top row). Next, D5_dia_anti substructure will form a 3-periodic crystal network ( Figure 29, bottom, left) while D5_dia_syn will arrange into a star-like quasi-crystal ( Figure 29, bottom, right).
The corresponding substructures of the hyper-graphenes in Figure 30 are illustrated in Figure 31 (top row). Alternating C20/C28 hyper-graphene domains with five-fold symmetry ( Figure 32) can result by sectioning a quasi-crystal (Figure 32, right) by an electron beam [103]. Pentagonal hyper-rings appearing in the core of these stars are illustrated in Figure 31, bottom.
Data for the above hyper-graphenes are collected in Table 19 [103]. Data for some small fullerenes and corresponding 5-fold and 6-fold hyper-cycles [104] are presented in Table 20.  A hyper-graphene could be conceived to appear when a thin layer of C60(Ih) is deposited on a (plane) surface. The polymerization process can start with a [2+2] cyclo-adduct but this is just the beginning of a more complex process, next following the coalescence of quasi-spherical units of C60(Ih) to form oligomers and finally a polymer (Figures 33 and 34); Table 21 supports this idea [104].   Let us detail the structures participating to such a process. Two dimers with joint face for C60(Ih) units can be designed (Figure 33, top): C60P2J5_115 (J5 meaning a pentagon identification) and C60P2J6_114 (J6 representing a hexagon identification). These two dimers have the total energy per C atoms comparable to C60(Ih); the HOMO-LUMO gap of "J5"-dimer is larger than that of "J6"-dimer (even "J5" dimer has no Kekulé structures).
Next, among the four trimers ( Figure 33, middle and bottom) the most stable (see the total energy per carbon atom and gap values in Table 22) appears to be C60P3J666_162. The two highly distorted trimers (C60P3J556_164 and C60P3J566_163) are less stable and further will not be considered.
The "J555" trimer C60P3J555_165 shows a lower gap probably because no Kekulé structure can be written. This could be not an argument since the "J5"-dimer also does not admit a Kekulé structure. At a higher number of carbon atoms (see structures in Figure 33) the Kekulé structures are possible for the both J-type polymers while the J6-type joining appear the most stable. It is no matter which one of the oligomers will be formed, the hyper-graphene has a good chance (see the boldface HL gap values in Table 21) to exist as areal structure. Note the hyper-graphene Le(Cor(C20))J5_165_1560 was designed by applying the leapfrog Le map operation [96][97][98] on the coronene-like structure made from the C20 smallest fullerene. The hyper-graphene Cor(C60)J6_162_1512 was designed by identifying the hyper-hexagons Hex(C60J6)_324.
Comparative computations, at HF, DFT and DFTB levels of theory, have been done on small substructures (Table 22). One can see that, in general, the ordering in the three approaches is preserved, of course with some exceptions. The main drawback of DFTB is the underestimation of the gap values in case of sp 2 carbon-only structures (see Table 22). However, DFTB is useful in ordering series of rather large carbon nanostructures [104].

Computational Details
The geometries of the polycyclic hydrocarbon molecules have been optimized at the HF/6-31G(d) and B3LYP/6-31G(d) level of theory, with the Gaussian 09 suite of programs [105]. The polybenzenes and fullerenes were optimized at the Hartree-Fock HF (HF/6-31G**) and DFT (B3LYP/6-311+G**) levels of theory, while the vibrational spectra (IR and Raman) were performed on the HF optimized structures. Sumanene fullerenes and carbon nanotube junctions were optimized at the Hartree-Fock HF (HF/6-31G**) level only.
Geometry optimization of the circulenes and cyclic compounds appearing in the isodesmic reaction schemes (benzene, naphthalene, cyclooctatetraene, indene, phenanthrene, azulene, acenaphthylene, fluorine and coronene) has been performed at HF/6-311 G(d,p) level of theory. No imaginary frequencies were obtained. In order to compute the enthalpies of formation of the 6-flowers [6:66], [6:(5,7)3] and [6:(5,6)3], the reaction energy for all the six isodesmic schemes was computed using the Equation derived from the Hess law: Also, the isodesmic and homodesmic reactions, using some experimental thermodynamic data, along with computed data (by DFT methods, e.g., B3LYP/6-311+G**) can bring a light on the stability of molecules, comparable with the experimental data [117]. B3LYP, Hartree-Fock HF and other method have been tested, in a comparative study on stability of fullerenes [118]. Magnetic susceptibility is also useful in this respect; values of exaltation of magnetic susceptibility for the basic polycyclic aromatic hydrocarbons (PAHs) have been collected in ref. [119]. Magnetic susceptibility can be calculated e.g., by B3LYP method, that properly indicate the trend of stabilization in PAHs [120].

Conclusions
In this review article, we presented computational arguments in supporting the possible existence of some not yet synthesized molecules, involved in nanosystems. Modeling small molecules enables one to build novel assemblies that may explain observed aggregates, this being not a trivial task of theorists. Four ideas have been detailed: Aromaticity of new fullerenes, patched with flowers of 6-and 8-membered rings, was discussed in terms of HOMA and NICS criteria. The spanned fullerenes patched by coronene and sumanene motifs evidenced clear electronic differences inside to outside the cage, as resulted from the NICS calculations. The calculated aromaticity parameters: HOMA, NICS, magnetic susceptibility, formation enthalpy, electronegativity, total hardness, electrophilicity and reactivity Fukui functions, computed on Hartree-Fock and DFT optimized molecular structures, provided a complex image on the electron distribution and stability of these yet hypothetical fullerenes, in agreement with the experimental data for the consisting patches, collected in the literature (as real molecules).
Polybenzene networks have been presented, from construction to energetic and vibrational spectra computations. The energetics and spectra of some repeating units, monomers, dimers, oligomers, involved in the construction of 3-periodic or 1-periodic polybenzene nanostructures, have been presented with the aim of helping experimentalists in eventual syntheses. The aromaticity of benzene patches (i.e., hexagonal rings) in polybenzenes seems to be rather close to that of the isolated benzene molecule, as suggested by their small distortions to the planarity. P-type crystal networks have been designed in several decorations; the reviewed data presented some zeolite-like 3-periodic nanostructures constructed with sumanene or derived patches, for which energetics and vibrational specta have been computed; data supported the idea that such "ordered schwarzites" could be real molecular crystals.
Construction and stability evaluation (at DFTB level of theory) of some exotic allotropes of diamond D5, involved in hyper-graphenes, was presented at the end of this review. Substructures of the hyper-diamond D5 was shown to form 3-periodic networks, as in D5-anti allotropes, or 1-periodic quasicrystal allotropes, as in D5-syn structures. Also, the D5-syn-allotrope can form hyper-graphenes, e.g., by cutting with an electron beam. In view of understanding the reliability of various computational approaches, comparative study was done using Hartree-Fock, Density Functionals and the semi-empirical DFTB method. Conclusion was that, in spite of some differences, the three approaches are useful in predicting the stability of substructures involved in nanosystems, DFTB being important in ordering some large atom number structures. Even the majority of the presented structures are yet hypothetical ones, they represent slides of a scientific dream.