# Atomic and Electronic Structure of Mini-Fullerenes: from Four to Twenty

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It is customary to assume that fullerenes are the carbon molecules having a shape close to a spherical surface which can be composed of regular pentagons combined with regular hexagons. Earlier we developed a method for calculating electronic and atomic structure for such fullerenes. Now we suppose that the restriction on fullerenes construction only from pentagons and hexagons is unnecessary. In other words, we take the term "fullerene" in a broad sense to mean any shape inscribed into to a spherical surface which can be composed of carbon atoms, each atom having three nearest neighbors, whenever discussing hollow carbon clusters. In addition to this definition, we suppose that not only atoms but also shared electron pairs, forming covalent bonds, are located on one and the same sphere. From this it follows that the geometry of both the electronic structure as well as of the atomic structure of fullerenes can be most conveniently constructed and studied with the help of spherical geometry. We enlarged here the approach developed earlier for usual fullerenes. The procedure created reveals the hidden symmetry of the shared electron pairs located on the same sphere as that of carbon atoms. The most interesting result is that we have found structural isomers of some fullerenes which differed in their surface electronic density.