Distributed curvature and stability of fullerenes.
Distributed curvature and stability of fullerenes.
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DOI：
10.1039/c5cp03643g
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发表时间：
2015-08
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影响因子：
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通讯作者：
P. Fowler;S. Nikolic;R. de los Reyes;Wendy J. Myrvold
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文献类型：
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作者：
P. Fowler;S. Nikolic;R. de los Reyes;Wendy J. Myrvold
非平面共轭π系统的能量通常用π稳定和与几何曲率相关的空间应变的平衡来定性描述。曲率也有一个纯粹的图理论描述：多面体图的一个顶点的组合曲率被定义为1减去顶点度的一半加上在该顶点相遇的面的倒数大小的总和。棱镜和反棱镜在每个顶点都有正的组合顶点曲率。排除这两个无限族，我们称任何其他处处有正组合曲率的多面体为PCC多面体。立方PCC多面体最初很常见，但随着顶点数的增加，最终会消失；目前构建的最大的例子有132个顶点。富勒烯具有具有n个顶点、12个五边形和（n/2 - 10）个六边形的立方多面体分子图。我们发现有39个PCC富勒烯，都在20≤n≤60的范围内。在这个范围内，PCC状态和稳定性之间只有部分相关，由最小五边形邻接定义。对于任何多面体，顶点曲率之和为2；对于富勒烯，顶点曲率的平方和与五边形邻接数线性相关，因此是较低（n≤60）富勒烯相对稳定性的直接度量。当n≥62时，具有最小五边形邻接数的非pcc富勒烯使均方曲率最小。当n≥70时，最小均方曲率意味着五边形的隔离，这是裸富勒烯稳定性的最强指标。
Energies of non-planar conjugated π systems are typically described qualitatively in terms of the balance of π stabilisation and the steric strain associated with geometric curvature. Curvature also has a purely graph-theoretical description: combinatorial curvature at a vertex of a polyhedral graph is defined as one minus half the vertex degree plus the sum of reciprocal sizes of the faces meeting at that vertex. Prisms and antiprisms have positive combinatorial vertex curvature at every vertex. Excluding these two infinite families, we call any other polyhedron with everywhere positive combinatorial curvature a PCC polyhedron. Cubic PCC polyhedra are initially common, but must eventually die out with increasing vertex count; the largest example constructed so far has 132 vertices. The fullerenes Cn have cubic polyhedral molecular graphs with n vertices, 12 pentagonal and (n/2 - 10) hexagonal faces. We show that there are exactly 39 PCC fullerenes, all in the range 20 ≤n≤ 60. In this range, there is only partial correlation between PCC status and stability as defined by minimum pentagon adjacency. The sum of vertex curvatures is 2 for any polyhedron; for fullerenes the sum of squared vertex curvatures is linearly related to the number of pentagon adjacencies and hence is a direct measure of relative stability of the lower (n≤ 60) fullerenes. For n≥ 62, non-PCC fullerenes with a minimum number of pentagon adjacencies minimise mean-square curvature. For n≥ 70, minimum mean-square curvature implies isolation of pentagons, which is the strongest indicator of stability for a bare fullerene.