The Character Table

Prior to interpreting the character table, it is necessary to explain the terms reducible and irreducible representations. We can illustrate these concepts using the NH3 molecule as an example. Ammonia belongs to the point group C3V and has six elements of symmetry. These are E (identity), two C3 axes (threefold axes of rotation) and three crv planes (vertical planes of symmetry) as shown in Fig. 1-22. If one performs operations corresponding to these symmetry elements on the three equivalent NH bonds, the results can be expressed mathematically by using 3x3 matrices.  

Consider the (clockwise rotation by 120°) operation, shown with its changes  

The square matrix on the right-hand side is called a representation for the symmetry operation, C%. For the o operation, we obtain 

These representations are called reducible representations since they can be block-diagonalized in the form 

The sum of the diagonal elements of a matrix is called the character (%) of the matrix. Hereafter, we use the term character rather than the representation since there is a one-to-one correspondence between them and since mathematical manipulation with x is simpler than with the representation. The characters of the reducible representations for the E, and o operations are 3, 0 and 1, respectively. The characters for C3 (counterclockwise rotation by 120°) is the same as that of C, and those for cr2 and cr3 are the same as that of crj. By grouping symmetry operations of the same character ( class ), we obtain 

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The characters of the irreducible representations of a synnnetry group are collected together into a character table and the character table of the group 3 is given in table A1.4.3. The construction of character tables for finite groups is treated in section 4.4 of [2] and section 3-4 of [3]. 

Table Al.4.4 The character table of a synmietry group for the molecule.

The character tables of these groups are given in table Al.4.6 and table Al.4.71. If there were no restriction on pemuitation symmetry we might think that die energy levels of the H2 molecule could be of any one of the following four syimnetry... 

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

Table Al.4.11 The character table of the molecular synnnetry group C2 (M)...

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. 

The last two colimms of the character table give the transfonnation properties of translations along the v, y. 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

In the sixth column of the main body of the character table is indicated the symmetry species of translations (7) of the molecule along and rotations (R) about the cartesian axes. In Figure 4.14 vectors attached to the nuclei of H2O represent these motions which are assigned to symmetry species by their behaviour under the operations C2 and n (xz). Figure 4.14(a) shows that... 

In the final column of the character table are given the assignments to symmetry species and These are the components of the symmetric polarizability tensor... 

The character tables for all important point groups, degenerate and non-degenerate, are given in Appendix A. 

Assign the allene molecule to a point group and use the character table to form the direct products A-2 x5i,5i X 82,82 xE and E X E. Show how the symmetry species of the point group to which 1,1-dilluoroallene belongs correlate with those of allene. 

For a symmetric rotor molecule such as methyl fluoride, a prolate symmetric rotor belonging to the C3 point group, in the zero-point level the vibrational selection mle in Equation (6.56) and the character table (Table A. 12 in Appendix A) show that only... 

As we proceed to molecules of higher symmetry the vibrational selection rules become more restrictive. A glance at the character table for the point group (Table A.41 in Appendix A) together with Equation (6.56) shows that, for regular tetrahedral molecules such as CH4, the only type of allowed infrared vibrational transition is... 

Equations (13-15) completely determine the character table of the symmetry group Q for a chiral nanotube. 

Hence we may conclude for a vibration to be active in the infrared spectrum it must have the same symmetry properties (i.e. transform in the same way) as, at least, one of x, y, or z. The transformation properties of these simple displacement vectors are easily determined and are usually given in character tables. Therefore, knowing the form of a normal vibration we may determine its symmetry by consulting the character table and then its infrared activity. 

The situation is the same for point X along the axis for 0, 7r/2a or tt/sl. The character table, time reversal properties, and basis functions are given in Table 12-5. The degeneracy in X2 is again absent in cases (2) and (4). 

Finally, this type of analysis can be carried out for any point in the Brillouin zone such that by using the transformation properties of spin waves and the character tables, one may obtain the spin-wave band structure throughout the zone. 

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. 

The tables of characters have the general form shown in Table 5. Each colipua represents a class of symmetry operation, while the rows designate the different irreducible representations. The entries in the table are simply the characters (traces) of the corresponding matrices. Two specific properties of the character tables will now be considered. 

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. 

The character table for 03h is seen to be composed of four submatrices,... 

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). 

E (for the identity) in Table 6 are accounted for. Furthermore, the totally symmetric representation is r(1) e A the latter notation is dial usually used by speetroscopists The construction of the remainder of the character table is accomplished by application of the orthogonality property of the characters [see Eq. (30) and problem 131. Standard character tables have been derived in this way for the more common groups, as given in Appendix VQI. 

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