5.3 Irreducible Representations and Character Tables

Irreducible representations, or symmetry species, are ways to depict all possible properties (such as molecular orbitals or normal vibrations) of a molecule in terms of their symmetry properties. The symmetry properties are defined in terms of the behaviour of the property when the symmetry elements of the molecular point group are applied. Thus, for instance, the s_CH-MO of methylene (Figure 5.12) is symmetrical with respect to rotation about the principal axis (it does not change):

Figure 5.12 Symmetrical behaviour of the methylene s_CH-MO with respect to rotation about the principal axis.

whereas the p_CH2-MO is antisymmetrical (it changes its phase, Figure 5.13):

Figure 5.13 Antisymmetrical behaviour of the methylene p_CH2-MO with respect to rotation about the principal axis.

All possible properties can be described in terms of their behaviour (symmetrical or antisymmetrical) with respect to the symmetry elements of the molecule. The combinations of these behavioural patterns are the irreducible representations. They are classified using a symbolic notation that defines at least some of their behaviours. The meanings of the indiividual characters in the names of the irreducible representations are:

Letters:

A Symmetrical with respect to a rotation about the principal axis

B Antisymmetrical with respect to a rotation about the principal axis

E Doubly degenerate

T Triply degenerate

Subscripts:

1 Usually means symmetrical with respect to reflection in a s_v plane, but the rules are more complicated if there are several different s_v planes

2 Usually means antisymmetrical with respect to reflection in a s_v plane, but the rules are more complicated if there are several different s_v planes

g (gerade) symmetrical with respect to an inversion centre

u (ungerade) antisymmetrical with respect to an inversion centre

Superscripts:

' symmetrical with respect to the single mirror plane in C_s, or to a s_h plane

" antisymmetrical with respect to the single mirror plane in C_s, or to a s_h plane

These rules are often more complicated, especially for degenerate point groups, which will not be discussed here, but give a general idea of the principles.

The characteristics of a point group and its irreducible representations are usually collected in a character table, such as the one shown in Table 5.1 for C_2v:

Table 5.1 Character table for the point group C_2v.

The name of the point group is given in the top left corner. The symmetry elements (in blue) are the column headers and the irreducible representations (in red) are given in the first column below the name of the point group. The characters (in green) are given in the body of the table and define whether the irreducible representation is symmetrical (+1) or antisymmetrical (-1) with respect to the symmetry element belonging to the column. Characters other than +1 and -1 can occur for degenerate point groups. The symbols to the right of the table give the irreducible representations of the molecular translation vectors (or dipole components) in the three Cartesian directions (x, y and z), those of the rotations about the Cartesian axes (R_x, R_y and R_z) and of the six independent components of the polarisabilty tensor (a_xx, a_yy, a_zz, a_xy, a_xz,and a_yz). These are important because, for instance, when we determine the normal vibrations of a molecule, we need to eliminate the rotations and translations. More importantly, however, vibrations with an irreducible representation corresponding to a translation vector are infrared active and those corresponding to one of the polarisability tensor components are Raman active. Thus, predictions about vibrational spectra can often be made simply from a knowledge of the irreducible representations of the normal vibrations. This and many other applications of symmetry are described in standard symmetry textbooks.

Symmetry Elements
Point Groups
Irreducible Representations and Character Tables
Next:
Degenerate Orbitals