Symmetry, axes elements

For an Abelian group, each element is in a class by itself, since X 1AX = AX IX = A. Since rotations about the same axis commute with each other, the group e is Abelian and has n classes, each class consisting of one symmetry operation. 

A, B are the hyperfine interactions expected for an electron in pure s and pure p orbitals respectively. The elements of the p-orbital hyperfine coupling tensors have cylindrical symmetry and the principal elements can be expressed aa A + B, A — B, A — B. These are obtained from the experimental tensor by use of the relationships (Ax + Ay + AJIZ = A, and Ax-A = B. 

The behavior of the matrix elements of the primitive functions with respect to symmetry operations of the helical chain is important in determining die character of the matrix elements in the first Brillouin zone. The -matrix element is completely symmetric, of course. Consider next, however, the matrix element, eq (F13). Gx-Ax can be rewritten as... 

Fig. 2-6. Form of the secular equation for linear molecules, using cartesian displacement coordinates. This illustrates the factoring made possible by the symmetry of the potential function. All nonvanishing elements fall in the shaded areas, and the three blocks correspond to the Ax s, Ay s, and the Az s, respectively.

Is substantial. By contrast, the fact that the valence AO s of A increase in energy in the order nd (n+l)s (n+l)p entitles these molecules to be formally "allowed" species, where the terms "forbidden" and "allowed" refer to the way in which the symmetries of the core and ligand MO s match. Since the nature of bonding within AX determines its stereochemistry, we can say that many differences between main group and transition element molecules which have yet to be discovered will be primarily due to the difference in core-ligand orbital matching as illustrated below, where the arrows indicate the way in which the orbital nodal planes increase ... 

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