Character tables, examples

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

Having assigned symmetry species to each of the six vibrations of formaldehyde shown in Worked example 4.1 in Chapter 4 (pages 90-91) use the appropriate character table to show which are allowed in (a) the infrared specttum and (b) the Raman specttum. In each case state the direction of the transition moment for the infrared-active vibrations and which component of the polarizability is involved for the Raman-active vibrations. 

Following the discovery of penicillins, an extensive program for the screening of culture fluids and residual mycelial material commenced which resulted in the discovery of a large number of pyrazinones and related 1-hydroxy-2-pyrazinones with pronounced antibiotic character. Some examples are shown in Table 4. One of the earliest substances to be isolated, aspergillic acid (110 = OH, = Me, R = Et, R = R = H, R = Pr ), was found... 

A formidable array of compounds of diverse structure that are toxic to invertebrates or vertebrates or both have been isolated from plants. They are predominately of lipophilic character. Some examples are given in Figure 1.1. Many of the compounds produced by plants known to be toxic to animals are described in Harborne and Baxter (1993) Harborne, Baxter, and Moss (1996) Frohne and Pfander (2006) D Mello, Duffus, and Duffus (1991) and Keeler and Tu (1983). The development of new pesticides using some of these compounds as models has been reviewed by Copping and Menn (2000), and Copping and Duke (2007). Information about the mode of action of some of them are given in Table 1.1, noting cases where human-made pesticides act in a similar way. 

They indicated that the softness parameter may reasonably be considered as a quantitative measure of the softness of metal ions and is consistent with the HSAB principle by Pearson (1963, 1968). Wood et al. (1987) have shown experimentally that the relative solubilities of the metals in H20-NaCl-C02 solutions from 200°C to 350°C are consistent with the HSAB principle in chloride-poor solutions, the soft ions Au" " and Ag+ prefer to combine with the soft bisulfide ligand the borderline ions Fe +, Zn +, Pb +, Sb + and Bi- + prefer water, hydroxyl, carbonate or bicarbonate ligands, and the extremely hard Mo + bonds only to the hard anions OH and. Tables 1.23 and 1.24 show the classification of metals and ligands according to the HSAB principle of Ahrland et al. (1958), Pearson (1963, 1968) (Table 1.23) and softness parameter of Yamada and Tanaka (1975) (Table 1.24). Compari.son of Table 1.22 with Tables 1.23 and 1.24 makes it evident that the metals associated with the gold-silver deposits have a relatively soft character, whereas those associated with the base-metal deposits have a relatively hard (or borderline) character. For example, metals that tend to form hard acids (Mn +, Ga +, In- +, Fe +, Sn " ", MoO +, WO " ", CO2) and borderline acids (Fe +, Zn +, Pb +, Sb +) are enriched in the base-metal deposits, whereas metals that tend to form soft acids... 

A further property of die dieter tables arises from the fact that every symmetry group has an irreducible representation that is invariant under all of die group operations. This irreducible representation is a one-by-one unit matrix (the number one) for every class of operation. Obviously, the characters, are all then equal to one. AS this irreducible representation is by convention taken to be the first row of all Character tables consists solely of ones. The significance of the character tables will become more apparent by consideration of an example. 

An example of the application of Eq. (47) is provided by the group < 3v whose symmetry operations are defined by Eqs. (18). If the same arbitrary function,

symmetry operation can be worked out, as shown in the last column of Table 13. With the use of the projection operator defined by Eq. (47) and the character table (Table 6), it is found (problem 16) that the coordinate z is totally symmetric (representation Ai). However, it is the sum xy + zx that is preserved in the doubly degenerate representation, E. It should not be surprising that the functions xy and zx are projected as the sum, because it was the sum of the diagonal elements (the trace) of the irreducible representation that was employed in each case in the... 

In effect, the division by two is the result of the molecular symmetry, as specified by the character table for the group 0. In general it is useful to define a symmetry number a (= 2 in this case), as shown below. The well-known example of the importance of nuclear spin is that of ortho- arid para-hydrogen (see Section 10.9.5). 

The group developed above to describe the symmetry of the ammonia molecule consisted only of the permutation operations. However, if the triangular pyramid corresponding to this structure is flattened, it becomes planer in me limit. The RF3 molecule shown in Fig. lb is an example of this symmetry. In this case it becomes possible to invert the coordinate perpendicular to the plane of the molecule, the z axis. Obviously, the operation of reflection in the (horizontal) plane of the molecule, <7h> is identical. It is easy, then, to identify the irreducible representations A and A" as symmetric or antisymmetric, respectively, under the coordinate inversion. The group composed of the identity and the inversion of the z axis is then <5 = s> whose character table is of the form of Table 7. 

The A and B labels in Table 1 follow the convention that A s have characters of +1 for the rotation axis of highest order (C2 in the present case) while B s have character -1. A, by convention, is the totally symmetric i.r., since all operations of the group turn something of A symmetry into itself. Every group has a totally symmetric i.r. I.r. s with suffix 1 are symmetric (character +1) under av, whereas those with suffix 2 are antisymmetric (character -1). Table 1 is an example of a character table. Two-dimensional representations are denoted by symbols E. 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

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< >. 

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. 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

Let us suppose a Z>3 local symmetry for the Eu + ions in a particular crystal. Following the same procedure as in Example 7.2, and using the character table... 

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

For several systems of complexes formed between soft acceptors and donors, the values of AHn are not only strongly negative, but approximately constant for several consecutive steps, i.e. just what we would expect if each hgand were consecutively coordinated by equivalent bonds of an essentially covalent character. Good examples are provided by the Hg2+ systems, including complexes formed with halides, pseudohahdes and sulfides, and also by the Cd2+ cyanide and Pd + chloride systems (Table 2). 

For example, the character table of C2v appears in reference books as follows ... 

Because the trace of a matrix is independent of the coordinate system, matrices representing operations that have the same effect in different coordinate systems must have the same trace. It is possible to use this fact to abbreviate the character tables. For example, consider the long and short versions of the character table of ... 

Consider for example the abbreviated form of the character table of Ds/ji... 

One application of character tables is the identification of the symmetry species of given objects. For example, what is the symmetry species of a displacement Az in the positive z direction in the symmetry group C2v, assuming that the C2 axis is along z7... 

The sequence of numbers arrived at constitutes the representation of the two Is orbitals with respect to symmetry. Such a combination of numbers is not to be found in the character table it is an example of a reducible representation. Its reduction to a sum of irreducible representations is, in this instance, a matter of realizing that the sum of the a,+ and gu+ characters is the representation of the two Is orbitals ... 

In the text, when the character of a set of orbitals is deduced to give a reducible representation, the reduction to a sum of irreducible representations has been carried out by inspection of the appropriate character table. In some instances this procedure can be lengthy and unreliable. The formal method can also be lengthy, but it is highly reliable, although not to be recommended for simple cases where inspection of the character table is usually sufficient. The formal method will be explained by doing an example. 

For example, the point group has three classes (and necessarily three irreducible representations) and its character table is shown in... 

Though the character tables for all the important point groups are readily available (see, for example, Appendix I at the end of this book), it makes a convenient summary of our results to see how the tables can normally be deduced without explicit knowledge of the matrices themselves. The following four rules can be used ... 

If we return to our equilateral triangle example and apply the above rules, we obtain the numbers in Table 9-6.1. Using this table in conjunction with the character table (Table 9-7.2) and the decomposition rule ... 

Character tables for many point groups are listed in Section 9.12 at the end of this chapter. As an example, consider the character table for 6D6/, the point group of benzene. This group is of order 24. The symmetry operations are found to be divided into the 12 classes (1) E (2) C6, C ... 

Consider the infrared vibrational transitions of C02 as an example. We first ask which vibrational levels can combine with the ground (00°0) level. The ground vibrational level always belongs to the totally symmetric representation, which is 2 for. The character table yields... 

Let us look at some examples. For S2v, (9.139) and the character table give as the characters of the direct-product representation A2 Bt... 

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