Trivalent polyhedron

Case (b) tells us that even monocycles have one Heilbronner mode and indeed, since n e) — n(v) + 1 is the number of rings in a polycyclic molecule, that any alleven polycyclic has as many Heilbronner modes as it has rings. Any all-even trivalent polyhedron such as the cube or 2n-gonal prism has as many Heilbronner modes as it has rings in its Schlegel diagram, i.e., one fewer than the number of faces of the polyhedron itself. 

Symmetry also adds useful detail to the picture for polyhedral 3D tt systems. A trivalent polyhedron obeys... 

We restrict attention to graphs G that have no pendant edges. For concreteness, take a graph G corresponding to a trivalent polyhedron P, where each edge has four... 

The Hiickel approximation considers only hopping between nearest neighbors, and hence the sum in this expression is limited to only those operators that link the starting site to its neighbours. We thus do not need all the irrep matrices, but only a few of them. As an example, the cube is a trivalent polyhedron, i.e., each atom has three bonds to its neighbors. The neighbors of site (a) can be reached by C4, C, where the twofold axis belongs to the bCj class. We thus need... 

Trivalent Polyhedra The dual of a deltahedron is a trivalent polyhedron, meaning that every vertex is connected to three nearest neighbours. The fullerene networks of carbon are usually trivalent polyhedra. This reflects the sp hybridization of carbon, which can form three specialized forms of the Euler symmetry theorem can be formulated. We may start from Eq. (6.138) and replace vertices by faces. The edge terms remain the same since they are totally symmetric under the local symmetries of the edges. Rotations of the edges by 90° will thus not affect these terms. 

Furthermore, by multiplying the vertices in a trivalent polyhedron by three, we have accounted for all the edges twice, since each edge is linked to two vertices, hence ... 

Fig. 17. Europium coordination polyhedra in Eu804 a) Trivalent Eu (1) polyhedron, b) Trivalent Eu (2) polyhedron, c) Divalent Eu (3) polyhedron...

An easy way of visualizing the structure of fullerenes is to make a physical model, for example, to assemble equal angle planar trivalent connectors and equal length plastic tubes. Mechanically, the polyhedron-like structure so obtained can be considered as a space frame with equal rigid nodes and equal elastic bars such that three bars meet and form angles of 120° at each node. The closed net shape arises by deformation of the bars in a state of self-stress. The edges of the polyhedron obtained... 

A simple way to appreciate the shape of fullerene is to construct a physical model in which rigid planar trivalent nodal connectors represent the atoms and flexible plastic bars (tubes) of circular cross-section represent the bonds. From a mechanical point of view the model may be considered as a polyhedron-like space frame whose equilibrium shape is due to self-stress caused by deformation of bars. We suppose that the bars are equal and straight in the rest position and that they are inclined relative to each other at every node with angle of 120°. The material of the bars is assumed to be perfectly elastic and that Hooke s law is valid. All the external loads and influences are neglected and only self-stress is taken into account. Then we pose the question What is the shape of the model subject to these conditions To answer this question we apply the idea used for coated vesicles by Tarnai Gaspar (1989). 

Figure 17. The variation of the RE-to-phosphate distances is shown as a function of the RE-ion radius for both the monazite- and xenotime-structure orthophosphate compounds. As shown in this figure, the shorter RE-to-phosphorous distances in the monoclinic structure compounds vary linearly with the trivalent RE ion radius with a slope that is close to 1. A similar variation is evident for the RE-P distances in the tetragonal xenotime-structure compounds—a trend that supports the comparison of the [001] polyhedron-P04 tetrahedron chain arrangement in the two structural types (after Ni et al. 1995).

The separation of trivalent actinides, such as and is of major concern in the field of nuclear technology. Den Auwer et al. (2000) studied the structure of N,N,N ,N -Tetraethylmalonamide (TEMA) complexes of lanthanides and americium by several methods including the X-ray absorption technique. They carried out first the X-ray diffraction study of the single crystals of Nd(N03)3(TEMA)2 and Yb(N03)3(TEMA)2 and compared the EXAFS spectra of these complexes in a solvent phase. The metal-centered polyhedron in the solvent phase was found to be similar to that of the solid-state complexes. They also measured an EXAFS spectrum of the Am(N03)3(TEMA)2 complex in the solvent phase and confirmed the similar coordination spheres with the Nd complex. 

The Sm atom has an 8-prismatic environment, the same as the trivalent U atom of U3S5, and the interatomic distances Sm-S and U(III)-S are equal. The U atom has the same 7-octahedral environment as the U(IV) atom in U3S5. As is usual in this kind of polyhedron, the two sulphur atoms which substitute at one apex of the octahedron are at a very short distance from one another (3.21 A). 

Mohapatra, P.K. and Khopkar, P.K. (1989) Hydrolysis of actinides and lanthanides hydrolysis of some trivalent actinide and lanthanide ions studied by extraction with thenoyltrifluoroacetone. Polyhedron, 16, 2071-2076. 

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