Octahedron symmetry

Fig. 3.8 The tetrahedron T symmetry), octahedron (Oh S5mmetry) and icosahedron (7h symmetry) possess four, six and twelve vertices respectively, and four, eight and twenty equilateral-triangular faces respectively.

Symmetry of orbitals on the B6 octahedron, (a) Six outward-pointing (sp) orbitals used for a bonding to 6 H. (b) Six inward-pointing (sp) orbitals used to form the fi framework bonding molecular orbital, (c) ComponenLS for one of the fiM framework bonding molecular orbitals — the other two molecular orbitals are in the yz and zr planes, (d) (Components ftM one of the t2 framework bonding molecular orbitals — the other two molecular orbitals are in the yz and x planes. 

Octahedral Having the symmetry of a regular octahedron. In an octahedral species, a central atom is surrounded by six other atoms, one above, one below, and four at the comers of a square, 176 complex in transitional metals, 418-420 geometric isomerism, 415 Octane number, 584... 

The compounds of the MMe205F type, where Me = Nb or Ta M = Rb, Cs, Tl, crystallize in cubic symmetry and correspond to a pyrochlore-type structure [235-237]. This structure can be obtained from a fluorite structure by replacing half of the calcium-containing cubic polyhedrons with oxyfluoride octahedrons. 

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

We shall shortly draw on both of these symmetry and energy aspects of Fig. 6-1 in the construction of molecular orbitals for the octahedron. First, however, let us extend the picture to molecules with more than two atoms. 

Hexaborides of a CaBg type are formed by K, the alkaline earths, Y and the larger lanthanides, as well as Th and some actinides ". The crystal structure of these compounds with cubic symmetry (Pm3m, O, ) (see Fig. 1) is characterized by a three-dimensional skeleton of Bg boron octahedra, the interstices of which are filled by metal atoms. The connection between two octahedra is by a B—B bond of length 1.66 X 10 pm, whereas the B—B bond lengths in one octahedron are 1.76 X 10 pm. ... 

Th = symmetry of an octahedron without fourfold axes [2/m3, short m3 ]. 

You say that your nonlinear molecule has the high symmetiy of a regular polyhedron, such as a tetrahedron, cube, octahedron, dodecahedron, icosahedron,... sphere. If it is a sphere, it is monatomic. On the other hand, if it is not monatomic, it has the symmetry of one of the Platonic solids (see the introduction to Chapter 8). 

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