Principal rotational axes

A continuous range of equisymmetric structures is permitted. Within that range, the other Archimedean polyhedra based on cubic geometry appear, in turn, as the local sets of vertices about the principal rotational axes are allowed to coalesce. 

The examination of coordinate transformations as local contractions and expansions of decorations about the poles of the principal rotational axes on the unit sphere for objects of Oh symmetry leads to intermediate geometries corresponding to particular Archimedean polyhedra related to the cube. In a similar manner, partial contractions and expansions of the decorations of the regular orbit of Ih point symmetry, i.e. the vertices of the great rhombicosidodecahedron, leads to the remaining polyhedra within the icosahedral family of Archimedean structures and orbits of Ih. 

The summary highlights an important observation, which identifies a general procedure that is particularly easy to apply for the cases of molecules in which a principal rotational axes can be identified [the dihedral groups]. For an orbit of n vertices, it is necessary only to identify the first occurrences of n distinct a group orbitals, which occur in the hierarchical order s-, p-, d-,. .. like, with respect to the central functions. Then the construction of the other group orbitals of the valence set decorations are formed by superimposition of the uj [jTe here] which operation leads to the Fz x Fa symmetry group orbitals, with the remainder identified on concerted local rotation of the uj into the here] and of symmetries... 

In this chapter, we extend our treatment of rotation in diatomic molecules to nonlinear polyatomic molecules. A traditional motivation for treating polyatomic rotations quantum mechanically is that they form a basis for experimental determination for bond lengths and bond angles in gas-phase molecules. Microwave spectroscopy, a prolific area in chemical physics since 1946, has provided the most accurate available equilibrium geometries for many polar molecules. A background in polyatomic rotations is also a prerequisite for understanding rotational fine structure in polyatomic vibrational spectra (Chapter 6). The shapes of rotational contours (i.e., unresolved rotational fine structure) in polyatomic electronic band spectra are sensitive to the relative orientations of the principal rotational axes and the electronic transition moment (Chapter 7). Rotational contour analysis has thus provided an invaluable means of assigning symmetries to the electronic states involved in such spectra. 

Figure 7.7 Rotational fine structure of the 0 absorption band in aniline vapor. The experimental rotational contour is shown at top the theoretical simulations labeled (b), (c), and (d) were generated using identical rotational constants, but assumed 0 transition moments polarized along theib, a. and c principal rotational axes, respectively. This work established that the S, - transition in aniline is polarized along the b axis (Fig. 7.8), or equivalently, y-polarized (cf. Fig. 7.5). Reproduced by permission from J. Christofferen, J. M. Hollas, and G. H. Kirby, Mol. Phys. 16, 441 (1969).

Figure 7.8 Orientations of principal rotational axes in aniline. The c axis is perpendicular to the paper.

Silicon carbide (carborundum, SiC) is of especial interest on account of its rich polymorphism, no fewer than six structures being known. As is to be expected, each carbon and silicon atom is tetra-hedrally co-ordinated by four atoms of the other kind, and two of the forms of carborundum have the zincblende and wurtzite structures. The close relationship between these two structures has already been discussed ( 4.13), and is emphasized by the many AX compounds (including ZnS itself) in which both are found. It is illustrated in fig. 8.03, where the cubic zincblende structure has been drawn with one of the cube diagonals vertical and parallel to the principal axis of the wurtzite structure. When viewed in this way it will be seen that both structures can be visualized as formed by the superposition of a series of puckered sheets of atoms, but that in zincblende successive sheets are identical (albeit translated) whereas in wurtzite they differ and are related by a rotation through 180° about the principal axis. In the two structures the sequence of sheets can therefore be symbolized as... 

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