Small‐sized cages of C_n are possible intermediate species for the formation of fullerene molecules in the discharge or laser vaporization reaction of graphite, but the studies on them are scattered mainly owing to their high reactivity and unavailability. Here, the coordinates of small‐sized carbon cages of C_n (n = 4‐30) are generated by our developed codes, and their stability and structural connectivity are studied using density functional theory. The interesting findings include the following: (a) Cage‐shaped isomers of C_n are favored with increasing size; (b) the ground state of cage‐shaped C₄, C₈, C₂₆, and C₃₀ are open‐shelled, and cage‐shaped D_3h‐C₆ is a second saddle point; (c) small‐sized cages welcome strained square(s); and (d) the most favored cages of C_n from C₁₆ to C₃₀ are structurally related to direct C₂ insertion without the rearrangement process.

中文翻译：

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小型碳笼的产生，稳定性和连通性

C _n的小笼可能是在石墨的放电或激光汽化反应中形成富勒烯分子的可能的中间物种，但是对它们的研究主要是由于它们的高反应性和不可利用性而分散的。在这里，由我们开发的代码生成了C _n（n = 4-30）小型碳笼的坐标，并使用密度泛函理论研究了它们的稳定性和结构连通性。有趣的发现包括：（a）C _n的笼形异构体随着大小的增加而受到青睐；（b）笼形C ₄，C ₈，C ₂₆和C _30的基态是开壳的，笼形D_3h- C ₆是第二个鞍点；（c）小型笼子，欢迎张紧的正方形；（d）从C ₁₆到C ₃₀的最喜欢的C _n笼在结构上与直接C ₂插入而不进行重排过程有关。