Structures of octahedral fullerenes and the application of discrete local curvature and continuum elastic theory to the study of fullerene's stability

There are two chapters in this dissertation. In Chapter 1, a modified elastic theory for $ ext{sp}^2$ pure carbon molecules has been proposed. The theory includes high order term in curvature, the bond stretching term and the tight-binding correction. The validity of this model has been examined by various graphitic-like molecules with different topology. The model can be applied on DFT (Density Functional Theory) optimized geometry or AIREBO (Adaptive Intermolecular Reactive Empirical Bond-Order) optimized geometry. Finally the most stable fullerene was found by constructing the candidates via the new model first and then verifying by VASP. In Chapter 2, a construction scheme of octahedral fullerenes has been built. After investigating the topological constraint, a fundamental polygon compatible with the octahedral symmetry was found. The fundamental polygon can be specified by four integers called index. However, the octahedral fullerene does not specified by a unique index and there is redundancy in the indexes which we called index symmetries. Besides symmetries corresponded to the geometrical symmetries of the graphene, there are symmetries originated from different dissection ways of the octahedral fullerenes. Finally all the possible orbits are clarified and an algorithm to eliminate these redundancy has been suggested.