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Advances in Physical Chemistry
Volume 2015, Article ID 506894, 6 pages
http://dx.doi.org/10.1155/2015/506894
Research Article

Geometry, Energy, and Some Electronic Properties of Carbon Polyprismanes: Ab Initio and Tight-Binding Study

1Department of Condensed Matter Physics, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow 115409, Russia
2Laboratory of Computational Design of Nanostructures, Nanodevices and Nanotechnologies, Research Institute for the Development of Scientific and Educational Potential of Youth, Aviatorov Street 14/55, Moscow 119620, Russia

Received 30 July 2015; Revised 7 October 2015; Accepted 11 October 2015

Academic Editor: Miquel Solà

Copyright © 2015 Konstantin P. Katin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We report geometry, energy, and some electronic properties of [n,4]- and [n,5]prismanes (polyprismanes): a special type of carbon nanotubes constructed from dehydrogenated cycloalkane C4- and C5-rings, respectively. Binding energies, interatomic bonds, and the energy gaps between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) have been calculated using density functional approach and nonorthogonal tight-binding model for the systems up to thirty layers. It is found that polyprismanes become more thermodynamically stable as their effective length increases. Moreover, they may possess semiconducting properties in the bulk limit.

1. Introduction

Carbon [n,m]prismanes (polyprismanes) can be regarded as stacked layers of dehydrogenated cycloalkane molecules, where is the number of vertices of the closed carbon ring and is the number of layers [1]. However, the polyprismanes’ ends are passivated by hydrogen atoms to avoid the dangling bonds. For large , polyprismanes (PP) represent single-walled carbon nanotube analogs with an extremely small cross-section as a regular polygon. Despite the fact that PP are of considerable fundamental and practical interest, their synthesis is very difficult task due to their high-strained carbon frame. Unlike carbon nanotubes, polyprismanes have rectangles on their surfaces instead of hexagons. So the angles between covalent C–C bonds are different from the value 109.5° usual for the sp3-hybridized carbon orbitals. Only “simplest” PP representatives are obtained: [2,3]prismane C6H6 (Ladenburg’s prismane) [2], [2,4]prismane C8H8 (hydrocarbon cubane) [3], and [2,5]prismane C10H10 [4]. Attempts to synthesize [2,m]prismanes starting from [2,6]prismane C12H12 failed. Nevertheless, hypothetical ways of synthesis based on high-level density functional calculations are predicted for [3,4]prismanes and [4,4]prismanes [1]. Moreover, structural characteristics and electronic properties of [n,m]prismanes with , [5]; , [6]; , [7, 8] are actively studied in terms of the density functional theory. The evolution of aromaticity along the thermally forbidden and dimerizations of cyclobutadiene and benzene to give [2,4]prismane and [2,6]prismane, respectively, is also discussed in the literature [9]. The special place is occupied by investigations of mechanical properties of polyprismanes, including the analysis of the edge doping (by replacing hydrogen atoms with chemical functional groups or radicals) influence on the PP mechanical characteristics [10, 11]. It is established that polyprismanes are molecular auxetics, that is, compounds having negative Poisson’s ratio [12]. In addition, high kinetic stability of small prismanes (namely, [2,6]prismanes and [2,8]prismanes) is also confirmed via direct molecular dynamics simulations [13]. Long [n,m]prismanes with the large values of are of special interest. They are considered as molecular wires for nanoelectronic applications [14]. In addition, recent ab initio calculations predicted that “long” (up to ) and even infinite () [n,4]prismanes were stable and could exhibit the metallic behavior [15]. Moreover, the particular electronic properties of long PP make them potentially suitable in the biomedical sector as biological ion carriers [1618]. So they can be used as drug delivery systems in the field of nanomedicine. But despite series of evident achievements in physical chemistry of long PP, their geometry, energy, and electronic properties are not well studied.

Nevertheless, PP are quite promising compounds. For example, polyprismanes can be functional nanowires with controlled electronic characteristics and can be used in computational logic elements. The incorporation of metastable carbon-nitrogen structures (nitrogen chains) inside the higher PP can stabilize the latter, which will make it possible to obtain an efficient high-energy material. The same principle of the formation of endohedral complexes (the encapsulation of therapeutic preparations inside polyprismanes) can be used for developing drug delivery systems. The use of PP as elements of measuring equipment, such as probes (tips) of the atomic-force microscopes, is also urgent. The absence of free covalent bonds makes polyprismanes insensitive to contaminants of the external medium (free radicals), which lowers the reaction ability of the tip making it possible to attain atomic resolutions in a microscope. All these possible applications pose quite concrete questions for the researchers: what are the limiting operating temperature modes of materials based on PP; how can we estimate lifetimes of the samples in extreme conditions (e.g., at cryogenic temperatures); how will electronic characteristics vary (e.g., conductivity) with an increase in the efficient sample length; how will edge doping affect their mechanical properties; and finally what compounds from the PP family are most thermodynamically and kinetically stable and the most probable candidates for the practical use?

In this paper, we present density functional theory and tight-binding calculations for the [n,4]- and [n,5]prismanes’ family with . The main purpose of this work is to obtain their binding energies, interatomic bonds, and HOMO-LUMO gaps and to analyze the behavior of these values in the bulk limit (). We hope that our study will stimulate the subsequent theoretical and experimental research of PP.

2. Materials and Methods

In our study we use density functional theory as well as the nonorthogonal tight-binding model (NTBM) [20]. Density functional theory (DFT) with Becke’s three-parameter hybrid method and the Lee-Yang-Parr exchange-correlation energy functional (B3LYP) [21, 22] with the electron basis set of 6-311G(d) [23] were used to optimize the geometries and obtain the structural, energy, and electronic (namely, HOMO-LUMO gaps) characteristics of the polyprismanes. The geometries of all polyprismanes were optimized using the GAMESS program package [24]. In the tight-binding model, the total potential energy of the system is the sum of the quantum-mechanical electronic energy and the “classical” ionic repulsive energy. The former is defined as the sum of one-electron energies over the occupied states, the energy spectrum being determined from the stationary Schrödinger equation (2S, 2Px, 2Py, and 2Pz orbitals of carbon atoms and 1S orbitals of hydrogen atoms are considered). The electronic Hamiltonian is calculated from overlap matrix elements using extended Hückel approximation [25] with Anderson distant dependence for Wolfsberg-Helmholtz parameter [26]. The overlap integrals are calculated using standard Slater-Koster-Roothaan procedure [27, 28]. Therefore, the analytical expressions for the overlap integrals greatly simplify the calculation of Hamiltonian. The parameters’ fitting of tight-binding model is based on the criterion of the best correspondence between the computed and experimental (not derived from ab initio calculations) values of binding energies, bond lengths, and valence angles of several selected small hydrocarbon molecules (for parameterisation details, see [20, 29]). To validate the model developed, we computed the binding energies, bond lengths, and valence angles for various H–C–N–O compounds not used in the fitting procedure and compared them with the experimental data from NIST database [19]. The resulting tight-binding potential is well suited to modeling various HkClNmOn molecules from small clusters to large systems as graphene, diamond, peptides, polyprismanes, and so forth [20]. For example, NTBM binding energies and structural properties for peptides such as bradykinin and colistin are in good agreement with the corresponding DFT calculations [20]. Unfortunately, we are not aware of experimental data derived for any polyprismane, except for cubane C8H8. For cubane, NTBM model is in excellent agreement with the existent experimental data. For example, experimental C–C and C–H bond lengths are 1.570 and 1.082 Å, respectively, and NTBM model gives 1.571 and 1.097 Å, respectively [30]. Binding energy obtained within the NTBM model (4.42 eV/atom) is also in good agreement with the experimental value (4.47 eV/atom) [30]. In addition, earlier we calculated the binding energies for hexa- and octaprismanes using NTBM and DFT/B3LYP/6-311G levels of theory. The binding energies of hexaprismane and octaprismane in terms of NTBM model were 4.56 and 4.47 eV/atom, respectively. These results agree well with our data found using the DFT/B3LYP/6-311G: 4.50 and 4.40 eV/atom for hexaprismane and octaprismane, respectively [13]. It should be noted that ab initio methods (say density functional theory) are very accurate but computer-resources demanding and hence applicable to small systems only. Calculation time within the DFT approach is estimated as , where is the number of atoms in molecular system. To check the convergence of the binding energies and the HOMO-LUMO gaps of “long” polyprismanes, we considered the molecular systems up to 500 carbon atoms. So the geometry and energy optimization of such systems in the frame of DFT is very computer resource-intensive or impossible. Therefore, the tight-binding approach is rather good compromise between the accuracy and computational costs and is commonly used for quantum chemistry modeling, including the calculation of binding energies, bond lengths, valence angles, and HOMO-LUMO gaps [3133].

The samples studied are [n,m]prismanes with m = 4 and 5 (tetra- and pentaprismanes, resp.) and (see Figure 1). To obtain the equilibrium structures of polyprismanes, we used the method of structural relaxation so that the corresponding initial configuration relaxed to a state with the local or global energy minimum under the influence of intramolecular forces only. First of all, the forces acting on the all atoms were calculated using the Hellmann-Feynman theorem [34]. Next, atoms were shifted in the direction of the forces obtained proportional to the corresponding forces. Then, the relaxation step was repeated. The whole structure is relaxed up to residual atomic forces smaller than  eV/Å.

Figure 1: Structures of the (a) [n,4]- and (b) [n,5]prismanes. Top-side view, bottom-top view. Symbols and correspond to interlayer and intralayer C–C bonds, respectively.

Note that for all polyprismanes studied we also calculated the frequency spectra in the framework of the same level of theory after the geometry optimization. All frequencies were positive for each metastable configuration.

3. Results and Discussion

3.1. Geometrical Parameters

For the all PP we studied it is possible to distinguish two different types of carbon–carbon covalent bonds (see Figure 1): parallel to polyprismane main axis interlayer C–C bonds () and perpendicular to main axis intralayer C–C bonds (). Their characteristic values for [n,4]- and [n,5]prismanes with and in the bulk limit are presented in Tables 1 and 2 for [n,4]- and [n,5]prismanes, respectively.

Table 1: Bond lengths of various prismanes obtained within the NTBM model. Experimental values for prismane (cubane) are given in parentheses [19]. All bond lengths are quoted in angstroms.
Table 2: Bond lengths of various []prismanes calculated within the NTBM model. Results obtained for []prismane (pentaprismane) within DFT/B3LYP/6-311G(d) are given in parentheses. All bond lengths are quoted in angstroms.

The ranges of bond lengths for [30,4]- and [30,5]prismanes are caused by the edge effects including C–H interactions. That is why the C–C bonds near the polyprismanes edge are longer than in the center. Optimized bond distances indicate that the interlayer C–C bonds in the “long” PP are shorter than the intralayer ones. This may be caused by the nonequivalence of p-orbitals spatial distribution of neighboring carbon atoms along and across the main axis of polyprismane. Thus, it leads to redistribution of electronic density, which defines the formation of two types of covalent bonds in polyprismane. It is interesting to notice that bond lengths in “simple” prismanes are quite longer than corresponding lengths in the bulk limit. As for bond lengths, the reverse trend is observed: C–C bonds in “simple” prismanes are shorter than corresponding ones in the bulk limit. Note that C–H bond lengths depend slightly on the effective length of PP.

3.2. Size Dependence of the Binding Energies and the HOMO-LUMO Gaps

The binding energies of [n,4]- and [n,5]prismanes ClHk are calculated by the equationwhere is the total number of atoms in the polyprismane, and are the energies of isolated carbon and hydrogen atoms, respectively, and is the total energy of the corresponding polyprismane. The polyprismane with higher binding energy (lower potential energy) is more thermodynamically stable and vice versa. HOMO-LUMO gap is defined as energy gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital. The binding energies and the HOMO-LUMO gaps obtained for small [n,4]prismanes () are summarized in Table 3.

Table 3: Binding energies and HOMO-LUMO gaps for various []prismanes calculated within the NTBM model and at DFT/B3LYP/6-311G(d) level of theory.

Figure 2 displays the binding energies and the HOMO-LUMO gaps of the [n,4]prismanes, for the larger .

Figure 2: Binding energies and HOMO-LUMO gaps versus the number of C4-rings obtained at the DFT/B3LYP/6-311G(d) (crosses) and NTBM (circles) levels of theory. The asterisk indicates that all HOMO-LUMO gaps obtained within the NTBM model are shifted to 0.35 eV.

Thus, we calculated and within the DFT approach for shorter PP () and using NTBM method for longer ones (). According to our calculations, NTBM model systematically underestimates the value of compared with DFT (see Table 3). It should be noted that tight-binding methods usually underestimate the value of the HOMO-LUMO gap of a few tenths of eV compared to DFT [35, 36]. Therefore, we shifted all HOMO-LUMO gaps obtained via the NTBM by 0.35 eV. From the overlapping area on Figure 2, one can see that the accuracy of the resulting NTBM is in good agreement with the DFT. Thus, the NTBM model is suitable to the calculation of the energy and electronic characteristics of PP.

For [n,4]prismanes, we found that and depend linearly on (here we do not take into consideration point with , which corresponds to the cubane C8H8 molecule due to its abnormal behavior; e.g., HOMO-LUMO gaps obtained within NTBM and DFT vary greatly). The corresponding analytical forms are

As evident from Figure 2, as the number of layers increases, becomes larger, reaching the “infinite” polyprismane eV/atom. On the other hand, decreases, reaching eV. Similar dependencies are observed for [n,5]prismanes (see Figure 3).

Figure 3: Binding energies and HOMO-LUMO gaps versus the number of C5-rings obtained within the NTBM model. All HOMO-LUMO gaps obtained are shifted to 0.35 eV.

Again and depend linearly on (again, we do not take into consideration point with , which corresponds to the pentaprismane C10H10 molecule). The analytic forms for [n,5]prismanes are

The increase of indicates that the [n,4]- and [n,5]prismanes become more thermodynamically stable as their effective lengths increase. In our opinion, the high thermodynamic stability of “long” PP is associated with the decreasing of boundary induced structural distortions as the number of layers increases. As for HOMO-LUMO gap, such behavior is typical for low-dimensional carbon nanostructures, for example, carbon nanotubes [37], graphene nanoribbons [38], and carbon nanoflakes [39]. Note that values for “infinite” PP are near to the upper limit of characteristic semiconducting value. In other words, these values are comparable with the corresponding band gaps for the wide-bandgap semiconductors, such as GaP (2.3 eV), GaN (3.4 eV), and ZnSe (2.7 eV) [40]. So, in the bulk limit for pristine PP, it is not possible to transport electrons, but additional doping or mechanical stresses may induce charge transfer.

4. Conclusions

In this paper we presented structural, energy, and some electronic properties of [n,4]- and [n,5]prismanes with . The data of numerical simulation obtained in this study indicate an increase of thermodynamic stability of PP as the number of layers increases. Binding energies estimated for both “infinite” tetra- and pentaprismanes are equal to 4.9 eV/atom. In contrast to binding energies, HOMO-LUMO gaps decrease as the effective length of PP increases. In the bulk limit () is equal to 2.4 and 3.2 eV for [n,4]- and [n,5]prismanes, respectively. These values are near to the upper limit of characteristic semiconducting value; therefore, it becomes possible to tune electronic properties of the PP by doping or mechanical stresses that will be useful for nanoelectronic applications.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The reported study was partially supported by RFBR, Research Project no. 15-32-20261 mol_a_ved. The useful comments of anonymous reviewers are gratefully acknowledged.

References

  1. E. G. Lewars, Modeling Marvels: Computational Anticipation of Novel Molecules, Springer, Dordrecht, The Netherlands, 2008.
  2. T. J. Katz and N. Acton, “Synthesis of prismane,” Journal of the American Chemical Society, vol. 95, no. 8, pp. 2738–2739, 1973. View at Publisher · View at Google Scholar · View at Scopus
  3. P. E. Eaton and T. W. Cole Jr., “Cubane,” Journal of the American Chemical Society, vol. 86, no. 15, pp. 3157–3158, 1964. View at Google Scholar · View at Scopus
  4. P. E. Eaton, Y. S. Or, and S. J. Branca, “Pentaprismane,” Journal of the American Chemical Society, vol. 103, no. 8, pp. 2134–2136, 1981. View at Publisher · View at Google Scholar · View at Scopus
  5. R. L. Disch and J. M. Schulman, “Ab initio heats of formation of medium-sized hydrocarbons. 7. The [n]prismanes,” Journal of the American Chemical Society, vol. 110, no. 7, pp. 2102–2105, 1988. View at Publisher · View at Google Scholar · View at Scopus
  6. R. M. Minyaev, V. I. Minkin, T. N. Gribanova, A. G. Starikov, and R. Hoffmann, “Poly[n]prismanes: a family of stable cage structures with half-planar carbon centers,” The Journal of Organic Chemistry, vol. 68, no. 22, pp. 8588–8594, 2003. View at Publisher · View at Google Scholar · View at Scopus
  7. S. Kuzmin and W. W. Duley, “Ab initio calculations of a new type of tubular carbon molecule based on multi-layered cyclic C6 structures,” Physics Letters A: General, Atomic and Solid State Physics, vol. 374, no. 11-12, pp. 1374–1378, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. S. Kuzmin and W. W. Duley, “Ab initio calculations of some electronic and vibrational properties of molecules based on multi-layered stacks of cyclic C6,” Fullerenes Nanotubes and Carbon Nanostructures, vol. 20, no. 8, pp. 730–736, 2012. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Alonso, J. Poater, and M. Solà, “Aromaticity changes along the reaction coordinate connecting the cyclobutadiene dimer to cubane and the benzene dimer to hexaprismane,” Structural Chemistry, vol. 18, no. 6, pp. 773–783, 2007. View at Publisher · View at Google Scholar · View at Scopus
  10. N. Pour, E. Altus, H. Basch, and S. Hoz, “The origin of the auxetic effect in prismanes: bowtie structure and the mechanical properties of biprismanes,” The Journal of Physical Chemistry C, vol. 113, no. 9, pp. 3467–3470, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. N. Pour, E. Altus, H. Basch, and S. Hoz, “Silicon vs carbon in prismanes: reversal of a mechanical property by fluorine substitution,” The Journal of Physical Chemistry C, vol. 114, no. 23, pp. 10386–10389, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. N. Pour, L. Itzhaki, B. Hoz, E. Altus, H. Basch, and S. Hoz, “Auxetics at the molecular level: a negative Poisson's ratio in molecular rods,” Angewandte Chemie, vol. 45, no. 36, pp. 5981–5983, 2006. View at Publisher · View at Google Scholar · View at Scopus
  13. S. A. Shostachenko, M. M. Maslov, V. S. Prudkovskii, and K. P. Katin, “Thermal stability of hexaprismane C12H12 and octaprismane C16H16,” Physics of the Solid State, vol. 57, no. 5, pp. 1023–1027, 2015. View at Publisher · View at Google Scholar
  14. S. L. Kuzmin and W. W. Duley, “Ab initio calculation of the electronic and vibrational properties of metal-organic molecules based on cyclic C6 intercalated with some group VIII transition metals,” Annalen der Physik, vol. 525, no. 4, pp. 297–308, 2013. View at Publisher · View at Google Scholar · View at Scopus
  15. P. A. S. Autreto, S. B. Legoas, M. Z. S. Flores, and D. S. Galvao, “Carbon nanotube with square cross-section: an ab initio investigation,” The Journal of Chemical Physics, vol. 133, no. 12, Article ID 124513, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Poater, A. G. Saliner, R. Carbó-Dorca et al., “Modeling the structure-property relationships of nanoneedles: a journey toward nanomedicine,” Journal of Computational Chemistry, vol. 30, no. 2, pp. 275–284, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Poater, A. G. Saliner, M. Solà, L. Cavallo, and A. P. Worth, “Computational methods to predict the reactivity of nanoparticles through structure-property relationships,” Expert Opinion on Drug Delivery, vol. 7, no. 3, pp. 295–305, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. A. Poater, A. G. Saliner, L. Cavallo, M. Poch, M. Solà, and A. P. Worth, “Tuning the electronic properties by width and length modifications of narrow-diameter carbon nanotubes for nanomedicine,” Current Medicinal Chemistry, vol. 19, no. 30, pp. 5219–5225, 2012. View at Publisher · View at Google Scholar · View at Scopus
  19. NIST Standard Reference Database 101, “Computational Chemistry Comparison and Benchmark DataBase,” 2015, http://cccbdb.nist.gov/.
  20. M. M. Maslov, A. I. Podlivaev, and K. P. Katin, “Nonorthogonal tight-binding model with H–C–N–O parameterisation,” Molecular Simulation, vol. 42, no. 4, pp. 305–311, 2016. View at Publisher · View at Google Scholar
  21. C. Lee, W. Yang, and R. G. Parr, “Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density,” Physical Review B, vol. 37, no. 2, pp. 785–789, 1988. View at Publisher · View at Google Scholar · View at Scopus
  22. A. D. Becke, “Density-functional thermochemistry. III. The role of exact exchange,” The Journal of Chemical Physics, vol. 98, no. 7, pp. 5648–5652, 1993. View at Publisher · View at Google Scholar · View at Scopus
  23. R. Krishnan, J. S. Binkley, R. Seeger, and J. A. Pople, “Self-consistent molecular orbital methods. XX. A basis set for correlated wave functions,” The Journal of Chemical Physics, vol. 72, no. 1, pp. 650–654, 1980. View at Publisher · View at Google Scholar · View at Scopus
  24. M. W. Schmidt, K. K. Baldridge, J. A. Boatz et al., “General atomic and molecular electronic structure system,” Journal of Computational Chemistry, vol. 14, no. 11, pp. 1347–1363, 1993. View at Publisher · View at Google Scholar · View at Scopus
  25. R. Hoffmann, “An extended Hückel theory. I. Hydrocarbons,” The Journal of Chemical Physics, vol. 39, no. 6, pp. 1397–1412, 1963. View at Publisher · View at Google Scholar · View at Scopus
  26. A. B. Anderson, “Derivation of the extended Hückel method with corrections: one electron molecular orbital theory for energy level and structure determinations,” The Journal of Chemical Physics, vol. 62, no. 3, pp. 1187–1188, 1975. View at Publisher · View at Google Scholar · View at Scopus
  27. J. C. Slater and G. F. Koster, “Simplified LCAO method for the periodic potential problem,” Physical Review, vol. 94, no. 6, pp. 1498–1524, 1954. View at Publisher · View at Google Scholar · View at Scopus
  28. C. C. J. Roothaan, “A study of two-center integrals useful in calculations on molecular structure. I,” The Journal of Chemical Physics, vol. 19, no. 12, pp. 1445–1458, 1951. View at Publisher · View at Google Scholar · View at Scopus
  29. K. P. Katin and M. M. Maslov, “Thermal stability of nitro derivatives of hydrocarbon cubane,” Russian Journal of Physical Chemistry B, vol. 5, no. 5, pp. 770–779, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. M. M. Maslov, D. A. Lobanov, A. I. Podlivaev, and L. A. Openov, “Thermal stability of cubane C8H8,” Physics of the Solid State, vol. 51, no. 3, pp. 645–648, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. M. B. Javan, “Electronic transport properties of linear nC20 (n5) oligomers: theoretical investigation,” Physica E: Low-Dimensional Systems and Nanostructures, vol. 67, pp. 135–142, 2015. View at Publisher · View at Google Scholar · View at Scopus
  32. W. Sukkabot, “Structural properties of SiC zinc-blende and wurtzite nanostructures: atomistic tight-binding theory,” Materials Science in Semiconductor Processing, vol. 40, pp. 117–122, 2015. View at Publisher · View at Google Scholar
  33. F. Bonabi and B. Bahrami, “Semi-empirical tight-binding method for calculating energy levels of hydrogenated silicon nanoclusters as a function of size,” Materials Science in Semiconductor Processing, vol. 27, no. 1, pp. 335–342, 2014. View at Publisher · View at Google Scholar · View at Scopus
  34. R. P. Feynman, “Forces in molecules,” Physical Review, vol. 56, no. 4, pp. 340–342, 1939. View at Publisher · View at Google Scholar · View at Scopus
  35. D. Lu, Y. Li, U. Ravaioli, and K. Schulten, “Empirical nanotube model for biological applications,” The Journal of Physical Chemistry B, vol. 109, no. 23, pp. 11461–11467, 2005. View at Publisher · View at Google Scholar · View at Scopus
  36. I. Prigogine and S. A. Rice, Eds., Advances in Chemical Physics: New Methods in Computational Quantum Mechanics, John Wiley & Sons, 1996.
  37. A. Rochefort, D. R. Salahub, and P. Avouris, “Effects of finite length on the electronic structure of carbon nanotubes,” Journal of Physical Chemistry B, vol. 103, no. 4, pp. 641–646, 1999. View at Publisher · View at Google Scholar · View at Scopus
  38. M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, “Energy band-gap engineering of graphene nanoribbons,” Physical Review Letters, vol. 98, no. 20, Article ID 206805, 2007. View at Publisher · View at Google Scholar · View at Scopus
  39. W. Hu, L. Lin, C. Yang, and J. Yang, “Electronic structure and aromaticity of large-scale hexagonal graphene nanoflakes,” Journal of Chemical Physics, vol. 141, no. 21, Article ID 214704, 2014. View at Publisher · View at Google Scholar · View at Scopus
  40. S. Kasap and P. Capper, Springer Handbook of Electronic and Photonic Materials, Springer, 2007.