Introduction

In the Nano-era, a period starting with the discovery of C60, in 1985, the carbon allotropes played a dominant role. Among the carbon structures, fullerenes (zero-dimensional), nanotubes (one dimensional), graphene (two dimensional), diamond, and spongy carbon (three dimensional) were the most studied [1–3], both from theoretical reasons and applications perspective.

Diamond D6, the beautiful classical diamond, with all-hexagonal rings of sp 3 carbon atoms (Fig. 1), crystallized in a face-centered cubic fcc network (space group Fd3m), has kept its leading interest among the carbon allotropes, even as the “nano” varieties [4–9]. Its aesthetical appeal and mechanical characteristics are of great importance in jewelry and technique. Synthetic diamonds are currently produced by a variety of methods, including high pressure–high temperature, chemical vapor deposition, ultrasound cavitation [10], etc.

Fig. 1
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Diamond D6 (left) and its repeating unit, adamantane (right)

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However, the diamond D6 is not unique: a hexagonal network called lonsdaleite (space group P63/mmc) [11], was discovered in a meteorite in the Canyon Diablo, Arizona, in 1967. Several diamond-like networks have also been proposed [2, 12, 13].

In a previous study, Diudea and Ilić [14] described some multi-tori (i.e., structures showing multiple hollows—see Ref. [15]), one of them illustrated in Fig. 2, left.

Fig. 2
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A multi-torus (left) and its reduced graph C17 (right), the seed of diamond D5

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The reduced graph of this multi-torus provided the structure for the seed of diamond D5: C17, (Fig. 2, right) consisting of a tetravalent atom surrounded by six pentagons, the maximum possible number of pentagons around a sp 3 carbon atom. According to the chemical nomenclature, C17 is a centrohexaquinane, a class of structures previously studied by Gund [16], Paquette [17], and more recently by Kuck [18–20].

Diamond D5 is the name given by Diudea to diamondoids consisting mostly of pentagonal rings [14, 21]. D5 is a hyperdiamond built up as a trinodal net in the frame of mtn structure; it belongs to the space group Fd-3m [22] while its seed is eventually the centrohexaquinane C17. However, D5 belongs to the family of Clathrates; it is precisely the type II clathrate C34 [23], of which Si34-analogue was already synthesized.

C17 can dimerize to 2 × C17 = C34 (Fig. 3), the repeating unit, in crystallographic terms, of the diamond D5 network. In crystallochemical terms, an adamantane-like structure, as ada_20_158 (Fig. 4, left) is the monomer which will probably condense to form the diamond D5 network (Fig. 4, right).

Fig. 3
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A joint of two C17 units (left) to give a dimer C34 (right), the repeating unit (in crystallographic terms) of the diamond D5 network

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Fig. 4
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Adamantane-like structure (left) of the diamond D5 network (right)

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In the above symbols, “20” refers to C20, which is the main unit of the hyper-diamond D5, while the last number counts the carbon atoms in structures.

The ratio C-sp 3/C-total trends to 1 in a large enough network. As the content of pentagons R [5] per total rings trends to 90% in this yet hypothetical carbon allotrope, the net was called the diamond D5 [21].

Method

The optimized structures, at Hartree–Fock HF (HF/6-31G**) and DFT (B3LYP/6-311+G**) levels of theory, were submitted to molecular dynamics MD procedure. All calculations were performed in gas phase by Gaussian 09 [24] while MD calculations were done in vacuum, using Amber 10.0 software [25]. The single point energy minima obtained for the investigated structures are shown in Table 1. Before the MD was run, the atomic charges were calculated according to Merz-Kollmann scheme via the RESP [26] procedure at HF/6-31G** level. The AMBER force field [27] was used for dynamic trajectory generation. There were several steps of molecular dynamics. After stabilization of energies and root mean square deviation (RMSD) values during run, the actual molecular dynamics were performed, in a cascade way. Each tested system was heated by 20 ps while MD simulations were 100 ns long. The visualizations were prepared in the GaussView program. After MD run, the values of RMSD and energies of analyzed structures were recorded: total energy (E tot), kinetic energy (E kin), and potential energy (E pot). In the analysis, averaged values of all generated points of energies and values of RMSD in every 1 ps of MD were used.

Table 1 The single point energies of the optimized structures at DFT (B3LYP/6-311+G**) level of theory
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Results and discussion

Stability evaluation was performed on four hypothetical seeds of D5, the all-carbon structure C17 (Fig. 2, right) and three trioxa-derivatives of C17. The isomer in Fig. 5, left was synthesized by Paquette and Vazeux [17], and is hereafter denoted P1. The other two structures, denoted D1 and D2 (Fig. 5, middle and right) were proposed, by Diudea, as possibly appearing in rearrangements of the Paquette’s P1 structure. The last two structures would be the appropriate ones in the next step of dimerization to C34, in fact the repeating unit [23] of the structure we call here D5.

Fig. 5
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C17_hexaquinane trioxo-derivatives: Paquette P1 (left) and Diudea, D1 (middle) and D2 (right)

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The stability of molecules was evaluated both in static and dynamic temperature conditions. The isomer D1 seems the most stable among all studied structures, as given by optimization in gas phase at DFT level (Table 1). In decreasing the stability, it follows P1 and D2. However, the all-carbon C17 appears the most stable at MD treatment, even at DFT level is the last one. This is probably because the C–C bond is more stable at temperature variations (see Fig. 8).

In MD, C17 keeps its structure up to about 1800 K, while its destruction starts at 2,000 K (Tables 2, 4; Figs. 6, 7). Kuck has reported a centrohexaindane, as the most symmetric structure in this series but also a benzo-centrohexaquinane [19, 20] as the last step structure in the synthesis of a non-planar 3D structure, designed according to mathematical rules. However, in the synthesis of centrohexaquinane-derivatives, C17 remained yet elusive.

Table 2 The averages total energy (E tot) values estimated, by MD, on geometries in the gas phase
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Fig. 6
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The plot of total energy (E tot) versus temperature (Temp)

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Fig. 7
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The plot of RMSD versus temperature (Temp)

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Very close to C17 behaves the oxygen-containing isomer D1, as expected from its highest stability at DFT level (Table 1-the values in bold).

Despite a very long time of molecular dynamics (100 ns) was leaded, it is believed that prolonged annealing at 1,800 K for both P1 and D1 isomers finally resulted in the destruction of these molecules. Thus, P1 and D1 isomers behave similarly in MD conditions. The isomer D2 was the least stable one, as the largest RMSD values were recorded for this isomer.

According to molecular dynamics, it is clear that by increasing temperature resulted in higher values of energy and RMSD (and their standard deviations) of all the analyzed structures, with high values of correlation. The plots of E tot versus temperature for all tested systems are given in (Fig. 6), while for RMSD the plots are given in Fig. 7. As expected, the correlations in the RMSD plot are a little lower than those for E tot. The MD calculations, listed in Tables 2, 3, and 4 show the followings.

Table 3 The values of standard deviations of E tot at a given temperature (see the center of each slide), for the four investigated structures
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Table 4 The averaged RMSD values estimated by molecular dynamics MD on the geometries in the gas phase
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As can be seen from Tables 2 and 3, the values of standard deviations of the averaged values of E tot are closely correlated with the values of temperature, in the range the molecular dynamics simulations were done. The values of these standard deviations at a given temperature are similar for all four studied structures (Tables 2, 3), due to their structural relatedness.

The smallest values of the RMS deviation are observed for C17, with the lowest values of standard deviation (δ) at all the studied values of temperature (Table 4).

In the case of P1, one can see a similar behavior but somewhat with larger values of RMSD (as well as values of the standard deviation), compared to the all-carbon structure C17 (Table 4). It confirms the structural stability of the above structures. The largest values of the RMS deviation were recorded for D2 isomer (Table 4; Fig. 7).

Visualization of the structural changes (first step destruction, the right column) is presented in Fig. 8.

Fig. 8
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The structure of the tested hypothetical seeds of the diamond D5 during molecular dynamics

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Conclusions

Structural stability of several hypothetical seeds of the diamond D5 was investigated. It was evaluated both in static and dynamic temperature conditions by molecular dynamics MD. During MD, the all-carbon C17 appeared the most resistant to changes of temperature. Structural and energetic stability of the other three seeds of D5 vary both with the values of temperature and evolution time in molecular dynamics and the arrangement of oxygen atoms in the molecules. Among all the studied structures, the D2 isomer is the most sensitive to changes in temperature. After optimization by B3LYP, D1 isomer seemed to be the most stable one. The structure stability of D1 and P1 isomers in MD are similar. These two isomers are only slightly more sensitive to temperature as compared with the all-carbon C17. These results could be useful in guiding further reactions, e.g., the dimerization to C34 and condensation to adamantane-like structures, finally leading to the diamond D5.