Point Groups and Their Character Tables

APPENDIX I. POINT GROUPS AND THEIR CHARACTER TABLES 

Infrared and Raman Spectra of Inorganic and Coordination Compounds, Sixth Edition, Part A Theory and Applications in Inorganic Chemistry, by Kazuo Nakamoto Copyright 2009 John Wiley Sons, Inc. 

Cp (or 5p) denotes that C (or Sp) operation is carried out successively n times. 

GtNERAL FORML LAS FOR CALCULATING THE NUMBER OF NORMAL VIBRATIONS IN EACH SPECIES 

These tables were quoted from G. Herzberg, Molecular Spectra and Molecular Structure Vol. II (Ref, j jp 

Point Group Total Number of Atoms Species Number of Vibrations  

NUMBER OF INFRARED- AND RAMAN-ACTIVE STRETCHING VIBRATIONS FOR MX Y -TYPE MOLECULES 

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The first two of the shapes are extremely common in chemistry, while the third shape is important in boron chemistry and many other cluster molecules (a cluster is defined as a molecule in which three or more identical atoms are bonded to each other) and ions. The three special shapes are associated with point groups and their character tables and are labelled, Td, Oh and Ih, respectively. The point group to which a molecule belongs may be decided by the answers to four main questions ... 

We will consider three examples of double groups and their character tables. The simplest of these is for the molecule with no spatial symmetry, which belongs to the point group Ci. Including spin symmetry and the E operation we get the C double... 

What has been mentioned up to now allows us to infer that the relevant information needed for a representation is given by the characters of its matrices. In fact, the full information for a given group is given by its character table. This table contains the character files of a particular set of representations the irreducible representations. Table 7.2 shows the character table of the Oh point group. A character table, such as Table 7.2, contains the irreducible representations (10 for the Oh group) and their characters, the classes (also 10 for the Oh group), and the set of basis functions. 

In this chapter, we first discuss the concept of symmetry and the identification of the point group of any given molecule. Then we present the rudiments of group theory, focusing mainly on the character tables of point groups and their use. 

In order to discuss the rest of the crystallographic point groups, one further has to consider the dihedral rotation groups D4 and D, and the cubic rotation group O. Their character tables, standard basis functions, and a useful choice of group generators are displayed in Tables 6,7, and 8. In this way the material required for symmetry considerations is directly available. 

F. A. Cotton, Chemical Applications of Group Theory, Third Edition, Wiley -Interscience, New York, 1990 M. Orchin, H. H. Jaffe, Symmetry. Point Groups, and Character Tables I, Symmetry Operations and Their Importance for Chemical Problems. J. Chem. Educ. 1970, 47, 372-377. 

Let us use the water molecule to illustrate the above statement. The normal modes of this molecule are shown in Figure 5-4. The point group is C2v, and the character table is given in Table 5-1. It is seen that all operations bring v and v2 into themselves so their characters will be ... 

If two or more atomic orbitals are interrelated under a symmetry operation of the point group and, accordingly, they together belong to an irreducible representation, their energies will also be the same. In other words, these orbitals are degenerate. Such atomic orbitals are parenthesized in the character tables. 

Some decades ago, Swalen and Costain(1959), Myers and Wilson (1960) have extended the use of the symmetry point groups for studying the double internal rotation in acetone [1-2]. Dreizler generalized their considerations to two Csv rotor molecules with frames of a lower symmetry than C2U, and deduced their character tables [3]. 

Assign the following molecules to point groups, look up their character tables, and indicate in each case whether one could expect doubly degenerate MOs. 

The character tables for the point groups which are relevant to most of the problems and discussions in this book are listed here. The transformation properties of the s, p and d orbitals are indicated by their usual algebraic descriptions appearing in the appropriate row of each table. 

We then discover an extremely important fact each normal coordinate belongs to one of the irreducible representations of the point group of the molecule concerned and is a part of a basis which can be used to produce that representation. Because of their relationship with the normal coordinates, the vibrational wavefunctions associated with the fundamental vibrational energy levels also behave in the same way. We are therefore able to classify both the normal coordinates and fundamental vibrational wavefunctions according to their symmetry species and to predict from the character tables the degeneracies and symmetry types which can, in principle, exist. 

Given the illustrations of the normal modes of a molecule, it is possible to identify their symmetry species from the character table. Each non-degenerate normal mode can be regarded as a basis, and the effects of all symmetry operations of the molecular point group on it are to be considered. For instance, the v mode of XeF4 is invariant to all symmetry operations, i.e. R(v 1) = (l)vj for all values of R. Since the characters are all equal to 1, v belongs to symmetry species Aig. For V2, the symmetry operation C4 leads to a character of -1, as shown below ... 

The character tables usually consist of four main areas (sometimes three if the last two are merged), as is seen in Table 4-5 for the C3v and in Table 4-7 for the C2h point group. The first area contains the symbol of the group (in the upper left corner) and the Mulliken symbols referring to the dimensionality of the representations and their relationship to various symmetry operations. The second area contains the classes of symmetry operations (in the upper row) and the characters of the irreducible representations of the group. 

It was discussed before that the irreducible representations can be produced from the reducible representations by suitable similarity transformations. Another important point is that the character of a matrix is not changed by any similarity transformation. From this it follows that the sum of the characters of the irreducible representations is equal to the character of the original reducible representation from which they are obtained. We have seen that for each symmetry operation the matrices of the irreducible representations stand along the diagonal of the matrix of the reducible representation, and the character is just the sum of the diagonal elements. When reducing a representation, the simplest way is to look for the combination of the irreducible representations of that group—that is, the sum of their characters in each class of the character table—that will produce the characters of the reducible representation. 

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i. e., to determine the irreducible representation of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (like O in H20 or N in NH3), the coordinate system can always be chosen in such a way that the central atom lies at the intersection of all symmetry elements of the group. Consequently, each atomic orbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitals will have the same symmetry properties as those basis functions in the third and fourth areas of the character table which are indicated in their subscripts. For all other atoms, so-called group orbitals or symmetry-adapted linear combinations (SALCs) must be formed from like orbitals. Several examples below will illustrate how this is done. 

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