Character tables, 279 construction

To what point group does CBr4 belong Using the appropriate character table, construct a reducible representation for the stretching modes of vibration. Show that this reduces to A + T2-... 

The characters of the irreducible representations of a synnnetry group are collected together into a character table and the character table of the group 3 is given in table A1.4.3. The construction of character tables for finite groups is treated in section 4.4 of [2] and section 3-4 of [3]. 

E (for the identity) in Table 6 are accounted for. Furthermore, the totally symmetric representation is r(1) e A the latter notation is dial usually used by speetroscopists The construction of the remainder of the character table is accomplished by application of the orthogonality property of the characters [see Eq. (30) and problem 131. Standard character tables have been derived in this way for the more common groups, as given in Appendix VQI. 

The possible wave functions for the molecular orbitals for molecules are those constructed from the irreducible representations of the groups giving the symmetry of the molecule. These are readily found in the character table for the appropriate point group. For water, which has the point group C2 , the character table (see Table 5.4) shows that only A1 A2, B1 and B2 representations occur for a molecule having C2 symmetry. 

As an example, to construct the character table for the Oh symmetry group we could apply the symmetry operations of the ABg center over a particularly suitable set of basis functions the orbital wavefunctions s, p, d,... of atom (ion) A. These orbitals are real functions (linear combinations of the imaginary atomic functions) and the electron density probability can be spatially represented. In such a way, it is easy to understand the effect of symmetry transformations over these atomic functions. 

At this point, we are able to construct the reducible representations D- of a group composed only of rotational elements. For instance, let us consider that the ion in the crystal has a symmetry group G = 0, whose character table (Table 7.4) consists of only rotational symmetry elements classes C . 

With these considerations in mind, the process of constructing the correct linear combinations of the subsets proceeds exactly as in the case of the carbocyclic systems. The correct coefficients of the atomic orbitals are simply the characters of the representations. For the E orbitals we will obtain some imaginary coefficients, but these may be eliminated by taking the appropriate linear combinations. We can thus write, almost by direct inspection of the character table of the C4 group ... 

This means that SALCs of , Tu and T2 symmetries can be constructed from the eight n orbitals of the B atoms. But, n MOs can be formed only if there are appropriate AOs on atom A with which they can interact. (Strictly speaking, a SALC on the B atoms alone is also a MO, but it is a nonbonding one.) Inspection of the Td character table shows that the following AOs are available. 

What atomic orbuals on carbon in the planar C03- anion could be used (on the basts of symmetry) to construct in-plane and out-of-plane n bonds First answer the question by thinking about the orientations of the orbitals relative to the geometry of the ion then answer it by using reducible representations and the appropriate character table. 

Let us now consider in some detail the molecular orbitals for an octahedral complex containing a first-row transition metal ion. The orbitals that will be used in the bonding scheme are the 3d, 4s, and 4p orbitals of the central atom and the ns and np orbitals of the ligands. The coordinate system that is convenient for the construction of a and n MO s is shown in Figure 8-2. The character table for the Ojj symmetry is given in Table 8-1. 

In otder to construct such sets of orbitals, it is most convenient to make use of group theory. Each set of equivalent directed valence orbitals has a characteristic symmetry group. If the operations of this group are performed on the orbitals, a representation, which is usually reducible, is generated. By means of the character table of the group7 this representation, which we shall call the a representation, can be reduced to its component irreducible representations. The s, p, and d orbitals of the atom also form representations of the group, and can also be divided into sets which form irreducible representations.8... 

The character table of C2v (Table 5) is easily constructed considering that in this group the barred and unbarred symmetry operations belong to the same class, with the exception of E and E, which always represent a class of their own. 

Both representations we constructed here are reducible since there are no 2- and 12-dimensional representations in the C2h character table (Table 4-7). The next question is how to reduce these representations. 

Important information about the symmetry aspects of point groups is summarized in character tables, described later in this chapter. To understand the construction and use of character tables, we first need to consider the properties of matrices, which are the basis for the tables. ... 

Figure 3.10 The consolidation of Figure 3.8 and Figure 3.9 to summarize the superposition procedure for the construction of cr and jt group orbitals on the vertices of the equilateral triangular orbit for the choice that the intrinsic symmetry, D31J, of the orbit is preserved. Note with reference to the Character Table for D31J that Fz is a and Fg is a. ...

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