Abstract
Very often the basic information about a nanostructure is a topological one. Based on this topological information, we have to determine the Descartes coordinates of the atoms. For fullerenes, nanotubes and nanotori, the topological coordinate method supplies the necessary information. With the help of the bi-lobal eigenvectors of the Laplacian matrix, the position of the atoms can be generated easily. This method fails, however, for nanotube junctions and coils and other nanostructures. We have found recently a matrix W which could generate the Descartes coordinates for fullerenes, nanotubes and nanotori and also for nanotube junctions and coils as well. Solving, namely, the eigenvalue problem of this matrix W, its eigenvectors with zero eigenvalue give the Descartes coordinates. There are nanostructures however, whose W matrices have more eigenvectors with zero eigenvalues than it is needed for determining the positions of the atoms in 3D space. In this chapter, we have studied this problem in the case of diamond structures. We have found that this extra degeneracy is due to the fact that the first and second neighbour interactions do not determine the geometry of the structure. It was found that including the third neighbour interaction as well, diamond structures were described properly.
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Acknowledgments
I. László thanks for the support of grants TAMOP-4.2.1/B-09/1/KONV-2010-0003, TAMOP-4.2.1/B-09/1/KMR-2010-0002 and for the support obtained in the frame work of bilateral agreement between the Croatian Academy of Science and Art and the Hungarian Academy of Sciences. The research of T. Pisanski has been financed by ARRS project P1-0294 and within the EUROCORES Programme EUROGIGA (project GReGAS N1-0011) of the European Science Foundation.
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László, I., Graovac, A., Pisanski, T. (2013). Drawing Diamond Structures with Eigenvectors. In: Diudea, M., Nagy, C. (eds) Diamond and Related Nanostructures. Carbon Materials: Chemistry and Physics, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6371-5_16
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DOI: https://doi.org/10.1007/978-94-007-6371-5_16
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