Vertex-split Icosahedron

The vertex-split Octahedron and the vertex-split Icosahedron are polycycles obtained from Octahedron and Icosahedron, respectively, by splitting a vertex into two vertices and the edges, incident to it, into two parts, accordingly. The vertex-split Octahedron is drawn on Figure 4.2,1 and both of them are given on Figure 8.3 they are the only, besides five Platonic r, q) — f, non-extensible finite (r, )-polycycles. 

A natural parameter to measure an (r, 4)-helicene, will be the degree of the corresponding homomorphism into r, q (on vertices, edges, and faces). For q > 4, helicenes appear with vertices, but not edges, having same homomorphic image. The vertex-split Octahedron is a unique such maximal helicene for (r, q) = (3,4) (two 2-valent vertices are such see Figure 4.2). There is a finite number of such helicenes for (r, q) = (3, 5) one of them is the vertex-split Icosahedron. 

Four exceptional non-extensible polycycles, depicted on Figure 8.3 are vertex-split Octahedron, vertex-split Icosahedron, and two infinite ones Pi x Pi = Prism<, and Tr% = APrismoo (see Section 4.2). 

For the parameters (r, q) = (3,5), (5,3), in Lemma 8.2.3, we will prove that the vertex-split Icosahedron is such a unique finite polycycle. ... 

Lemma 8.23 ([DSS06]) All finite elliptic non-extensible (r, q)-polycycles are two vertex-splittings (of Octahedron and Icosahedron see first two in Figure 8.3) and five Platonic r, q) (with a face deleted). 

The essential band gap is 1.50 for c and 1.46 eV for E-i-c, and the upper valence band with a distance of 1.32 and 1.29 eV, respectively, from the conduction band is identified as the split-off valence band caused by a static Jahn-Teller effect in the icosahedron as described in Sec. II.A. It was shown that this effect acts on the icosahedra at the vertex of the unit cell only, and accordingly the density of states of the band corresponds to the number of rhombohedral unit cells (N = 1.35 X 10 cm ) and is comparably low (56,57). This is important for the electronic transport mechanism described in the following. 

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