Energy Levels and Orbitals of the Simplest Clusters in N-Dimensions1

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We investigate the energy levels and orbitals, of the simplest clusters in arbitrary n-dimensions wherein each atom is attached to all of the others, and there is but one orbital per atom. This was done using a model analogous to Hückel molecular orbital theory wherein we naturally obtained the adjacency matrix and directly solved for the eigenvalues in simple closed form. Symmetry-based reasoning allows for facile solution, and also description, of the corresponding eigenvectors as well. We also discuss the related situation where there is a central atom symmetrically attached in n dimensions to n + 1 hydrogens.

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