Character tables of point groups

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. 

Character tables of point groups of high symmetry (n > 2) have entries other than 1, but we need not deal with such groups here. 

For a given molecule belonging to a particular point group, it is possible to consider the various symmetry species as indicating the behavior of the molecule under symmetry operations. As will be shown later, these species also determine the ways in which the atomic orbitals can combine to produce molecular orbitals because the combinations of atomic orbitals must satisfy the character table of the group. We need to give some meaning that is related to molecular structure for the species A1( B, and so on. 

It has turned out that the most concise description of the symmetry species compatible with a molecular point group, that still includes enough iirformation for useful predictions, is the group character table. The character table of a group is a list of the traces of sets of matrices that form groups isomorphic to the group or to one of its subgroups. 

Symmetry selection rules for Raman spectrum can be derived by using a procedure similar to that for the IR spectrum. One should note, however, that the symmetry property of symmetry species of six components of polarizability are readily found in character tables. In point group C3V, for example, normal vibrations of the NH3 molecule (2A1 and 2E) are Raman-active. More generally, the vibration is Raman-active if the component(s) of the polarizability belong(s) to the same symmetry species as that of the vibration. 

An isolated n-atom molecule has 3n degrees of freedom and in—6 vibration degrees of freedom. The collective motions of atoms, moving with the same frequency and which in phase with all other atoms, give rise to normal modes of vibration. In principle, the determination of the form of normal modes for any molecule requires the solution of equation of motion appropriate to the n-symmetry. Methods of group theory are important in deriving the symmetry properties of the normal modes. With the aid of the character tables for point groups and the symmetry properties of the normal modes, the selection rules for Raman and IR activity can be derived. For a molecule with a center of symmetry, e.g. AXe, octahedral molecule, a non-Raman active mode is also IR active, whereas for the BX4 tetrahedral molecule, some modes are simultaneously IR and Raman active. 

TABLE 6.3 Selected character tables for point groups of non-linear molecules. More tables appear in the Appendix. 

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. 

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. 

A planar molecule of point group 03b is shown in Fig. 5. The sigma orbitals i, <72 and (73 represented there will be taken as the basis set Application of the method developed in Section 8.9 yields the characters of the reducible representation given in Table 14. With the use of the magic formula (Eq. (37)] the structure of the reduced representation is of the form Ta — A, ... 

The following conventions are used to label species in the character tables of the various point groups ... 

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. 

So far, we have seen how to use the character tables of the point symmetry groups to interpret the optical spectra of some ions. When dealing with spin functions of ions, we have the possibility of half-odd integer values for the spin (or for the... 

Consider an ion with one 3d electron situated in a cubic environment such as a Mg++ site in MgO. The symmetry transformations of this environment constitute the point group Oh, the character table of which is given in Table I. 0 contains the following classes of elements ... 

Q Look up the transformation properties of the 2s and 2p orbitals of the nitrogen atom in the character tables of the C3v point group in Appendix 1 to confirm the content of Table 6.1. Carry out the procedure for classifying the Is orbitals of the three hydrogen atoms as group orbitals in the pyramidal molecule. 

Since we will continually be requiring the characters of the irreducible representations of the point groups, it is convenient to put them together in tables known as character tables- In the character table of a point group each row refers to a particular irreducible representation and, since the characters of operations of the same class are identical, only a single entry (C,) is made for all the operations of a given class. The columns are headed by a representative element from each class preceded by the number of elements or operations in that class gf. 

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