Degenerate vertex

The initial point could be an interior infeasible point it is particularly useful in two types of circumstance first, when we want to exploit a point as a first attempt resulting from the solution of a problem that is slightly different from the original one second, when the starting point is a degenerate vertex, we can slightly move away from this point. 

The first term in (13), also called the diagonal term (Berry 1985), originates from periodic orbit pairs (p,p ) related through cyclic permutations of the vertex symbol code. There are typically n orbits of that kind and all these orbits have the same amplitude A and phase L. The corresponding periodic orbit pair contributions is (in general) g n - times degenerate where n is the length of the orbit and g is a symmetry factor (g = 2 for time reversal symmetry). 

While Hirsch conceived his 2(n + l)2 electron rule for spherical aromatics, subsets of three-dimensionally aromatic molecules having very high symmetries ( Ti, Oj, h, etc.), it can be applied to lower symmetry clusters such as the nine-vertex examples above. In cluster molecules the highest degeneracy MOs of a spherically harmonic atom set split into related, but lower degeneracy (or even non-degenerate) components. 

We may also use the TSH formalism to explain systematic deviations from the usual deltahedral electron count. Such cases arise when the two members of a pair of degenerate orbitals are paired with one another, and must therefore both be nonbonding. This will be the case in any cluster where F / contains an odd number of fi-type irreducible representations (IR s). Fowler proved that any cluster with a rotation axis of order 3 or more, and a single vertex atom lying on that axis, would be forced to deviate from the usual skeletal electron count. These results were generalized by Johnston and Mingos, who classified clusters as nonpolar, polar, or bipolar according to the number of atoms on the principal rotation axis, that is, 0, 1, or 2, respectively. For... 

Note that p exits an F vertex (i.e., p represents electron creation or hole annihilation), and q enters an F vertex (i.e., q represents electron annihilation or hole creation). Also, p and q exit V vertices, and r and s enter but p and q enter T vertices, while r and s exit. Since V and T vertices are degenerate, the assignment of values to these vertices is determined only up to phase [cf. Eqs. (85)—(87)]. It will expedite the establishment of the overall phase factor of the matrix element (cf. Rule 3 later) if antisymmetrized component matrix elements with the same ordering of indices as the corresponding direct matrix elements are used. Hence, p and r, q and s, etc., are associated or considered participants in the same basic interaction. 

Retardation effects. If the phonon contribution dominates the bare vertex (6), the retardation effects associated with heavy ions can play an important role in the many-body theory. In order to develop this point in greater detail let us, for reasons of clarity, ignore the Coulomb contribution to the bare vertex (6). Some simple vertex corrections are shown in Fig. 1. These particular diagrams are chosen because in the one-dimensional case (ij = r/ — 0) they all yield the same > g6log2T contribution to the vertex, provided that the retardation effects are neglected. Such a degenerate situation is usually named parquet. 

Figure 2. The Kj 3 bipartite graph as a topological representation of the degenerate planar isomerization of a tetrahedron (Tj) to its enantiomer through a square planar intermediate (O4/,). The isomers corresponding to the vertices of the K22 bipartite graph are depicted next to the vertex labels.

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