T , character table

The trivinylmethyl radical C(CH CH,)3 has seven carbon atoms and seven 77-electrons and belongs to the 3h point group. We will however use the lower symmetry point group to which the molecule also belongs. The labeling of the carbon atoms is shown in Fig. 10-7.1. In Table 10-7.1 we show how the seven 2p, atomic orbitals 2,... 7) transform under the operators 0M and from these results we obtain the characters of rA0 they are given together with the < t character table in Table 10-7.2. It will be noticed that the Ts representation has been... 

The H2O molecule, therefore, has three normal vibrations, which are illustrated in Figure 4.15 in which the vectors attached to the nuclei indicate the directions and relative magnitudes of the motions. Using the C2 character table the wave functions ij/ for each can easily be assigned to symmetry species. The characters of the three vibrations under the operations C2 and (t (xz) are respectively + 1 and +1 for Vj, - - 1 and + 1 for V2, and —1 and —1 for V3. Therefore... 

Inspection of this character table, given in Table A. 12 in Appendix A, shows two obvious differences from a character table for any non-degenerate point group. The first is the grouping together of all elements of the same class, namely C3 and C as 2C3, and (t , and 0-" as 3o- . 

A molecule has a permanent dipole moment if any of the symmetry species of the translations and/or T( and/or 1/ is totally symmetric. Using the appropriate character table apply this principle to each of these molecules and indicate the direction of any non-zero dipole moment. 

Acetylene (HC=CH) belongs to the point group whose character table is given in Table A.37 in Appendix A, and its vibrations are illustrated in Figure 6.20. Since V3 is a vibration and T T ) = 2"+, the 3q transition is allowed and the transition moment is polarized along the z axis. Similarly, since Vj is a vibration, the 5q transition is allowed with the transition moment in the xy plane. 

For readers unfamiliar with these techniques, it might be helpful at this point to work out an example in some detail. We choose that of the allene skeleton, already discussed somewhat in this section, and at first we limit ourselves to achiral ligands, so that G = S4. The character table for S4 is shown in Table 1. In this case, the subgroup is just D2a, and its rotational subgroup is D2. Table 2 shows the classes of T>za, the number of elements in each, the class of S4 and of S4 to which each belongs, and the character of each for the representation, T< >. 

Table 7.4 The character tables of group O and its subgroup >4. The irreducible representation T] of group O appears written below as a reducible representation in >4...

In the ideal case of free Eu + ions, we first must observe that the components of the electric dipole moment, e x, y, z), belong to the irreducible representation in the full rotation group. This can be seen, for instance, from the character table of group 0 (Table 7.4), where the dipole moment operator transforms as the T representation, which corresponds to in the full rotation group (Table 7.5). Since Z)° x Z) = Z) only the Dq -> Fi transition would be allowed at electric dipole order. This is, of course, the well known selection rule A.I = 0, 1 (except for / = 0 / = 0) from quantum mechanics. Thus, the emission spectrum of free Eu + ions would consist of a single Dq Ei transition, as indicated by an arrow in Figure 7.7 and sketched in Figure 7.8. 

Reduction of T using the C2v character table shows that ... 

Given the characters x of a reducible representation T of the indicated point group 9 for the various classes of 9 in the order in which these classes appear in the character table, find the number of times each irreducible representation occurs in T. 

The character table for the point group of HsO SfST is given in Table 9-7.1 and below it we show the characters for the T° representation (found from eqn (9-6.1)). From these characters we obtain... 

This information is sufficient to classify the MOs with respect to the irreducible representations of and using the character table, we see that Of, Of, and Of belong to T and Of, Of, Of, and Of belong to... 

If the spin-orbit coupling is relatively small, we might wish to consider first the splitting of the 2D state by the environment and then the further splitting of the resulting states by spin-orbit coupling. To do this we first use the Z)4/l character table to find that the 2D state splits into A,g, B2 t, and En. These... 

The character table shows that the rotations transform as T, and the trans-... 

For the pure rotation group /, the outlined section in the upper left is the character table the g subscripts should, of course, be dropped and (a y, z) assigned to the T, representation. 

The atomic orbitals suitable for combination into hybrid orbitals in a given molecule or ion will he those that meet certain symmetry criteria. The relevant symmetry properties of orbitals can be extracted from character tables by simple inspection. We have already pointed out (page 60) that the p, orbital transforms in a particular point group in the same manner as an x vector. In other words, a px orbital can serve as a basis function for any irreducible representation that has "x" listed among its basis functions in a character table. Likewise, the pr and p. orbitals transform as y and vectors. The d orbitals—d d dy, d >, t, and d ,—transform as the binary products xy, xz, yr, x2 — y2, and z2, respectively. Recall that degenerate groups of vectors, orbitals, etc, are denoted in character tables by inclusion within parentheses. 

Fig. 15.3 Identification of the synwietnes of the C—O stretching vibrations (represented as vectors) for Fe(CO)j. The reducible representation. T, is derived by counting the number of vectors remaining unmoved during each operation of the point group (see Appendix D for character table). Its irreducible components are obtained by application of Eg. 3.1...

Using the character tables we can reduce x° to A E T2, while x1 is the character of the irrep T. Hence the occupation f2 gives lA, 1E, 1T2, and ZT electronic states. (For those who choose to explore this matter in greater detail, we note that the case we have just considered is tabulated incorrectly by Hurley [17].)... 

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