Nanostructures and Eigenvectors of Matrices

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 not only of fullerenes, nanotubes, and nanotori but also of nanotube junctions and coils. 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 such cases the geometry of nanostructure can be obtained with the help of a projection from a higher-dimensional space in a similar way as the quasicrystals are obtained.