Characters from orbits permutation

Figure 1.7 The worksheet for the calculation of the permutation character and its direct sum components, listed as Mulliken symbols from orbit lists. This display is accessed from the Characters from Orbits command button of the window shown in Figure 1.5.

Suppose that G is the group of symmetry operations of a polyhedron or polygon, with vertices corresponding to the atomic positions in a particular molecular structure. The division of the structure into orbits, as sets of vertices equivalent under the actions of the group symmetry operations and the calculation of associated permutation representations/characters were described in Chapter 2. In this chapter, the identity between the permutation representa-tion/character on the labels of the vertices of an orbit and the a representation/character on sets of local s-orbitals or a-oriented local functions is exploited to constmct the characters of the representations that follow from the transformation properties of higher order local functions. 

For the Fe3(CO)i2 example, a full analysis requires the determination of tt and 8 characters of the Fes orbit and the four sets of n characters arising from the 2p atomic orbitals of the two sets of 06v and Oen orbits of the Dsh point symmetry molecular structure, shown in Figure 3.4, in addition to the individual permutation characters over the vertices of the orbits. 

In electronic structure problems, we would normally be interested in bonding interactions of s, p and d-atomic orbitals from atoms sited on the vertices of a molecular stmcture and hence only Fcr, F r and F. In vibrational problems, the mechanical representation is Fcoordinates = Fff X Fxyz = Fcr + F r for an empty cluster and F + F r + Fxyz for a cluster with a centrally placed atom. For the cases of interest in molecular problems, equations 3.5 to 3.7, which follow from a knowledge of the permutation characters alone, can be used to generate, once and for all, the foil set of reducible characters for the orbits of the molecular point groups. 

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