八面體富勒烯的結構探討與局部曲率及彈性理論在富勒烯穩定性的應用

Abstract

本篇論文分成兩大章，第一章提出了針對sp2純碳分子的彈性模型。除了一般的彈性項，我們加入了高階項、鍵長修正項及緊縛模型項，以期達到更好的預測能力。我們用多個具有不同拓撲特徵的類石墨烯分子，來驗證此模型的可信度，皆得到了不錯的結果。此外，本模型除了密度泛函理論優化結構外，也適用於一些半經驗方法的優化結構。最後我們將此模型用來建構最穩定的C102~C140籠狀富勒烯的預測名單，再以VASP軟體做確認。 第二章則描述了八面體對稱富勒烯的建構方式。在研究其拓撲限制後，我們提出了符合八面體對稱的基本多邊形，並以四個整數組成索引來對應之，並用剪貼方式組合出八面體對稱富勒烯。在進一步研究其組合方式後，我們發現索引的對稱，除了來自於石墨烯的簡單幾何對稱外，還有來自於不同切割方式的切割對稱；並針對於此提出了消去所有對稱以達到一一對應的演算法。

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.