1 2 character table

TABLE 8.6 The character table containing the complete list of irreducible representations (IRRs) for the point group, along with their Mulliken symbol names. 

6 are both diagonal matrices). The 9x9 matrix representing the identity operation in the representation for the H2O molecule is an example of a diagonal matrix. 

Using trigonometry, the following statements will be true (where r is the length of the basis vector)  

Rotation of a vector having length r around the z-axis by an angle a. 

the 2x2 matrix that converts the vector (X, / ) into the vector (xj, yj) is therefore given by Equation (8.14)  

Some point groups, viz. the cyclic groups C , C(2/j+i), S2n, and also T and T, have irreps with complex characters. In these cases, for an irrep 71 with complex characters, there will always be a complementary irrep with a complex-conjugate character string, which is denoted as 71. Hence, one has 

In the next chapter, we will present various chemical applications of group theory, including molecular orbital and hybridization theories, spectroscopic selection rules, and molecular vibrations. Before proceeding to these topics, we first need to introduce the character tables of symmetry groups. It should be emphasized that the following treatment is in no way mathematically rigorous. Rather, the presentation is example- and application-oriented. 

As shown in Table 6.3.1, we can see that the C2v character table is divided into four parts Areas I to IV. We now discuss these four areas one by one. 

Additionally, it is noted that, mathematically, each irreducible representation is a square matrix and the character of the representation is the sum of the diagonal matrix elements. In the simple example of the C2v character table, all the irreducible representations are one-dimensional i.e., the characters are simply the lone elements of the matrices. For one-dimensional representations, the character for operation R, x(R), is either 1 or -1. 

Normal vibrational modes of the water molecule v (symmetric stretch) and V2 (bending) have A symmetry, while V3 (asymmetric stretch) has 2 symmetry. 

Before leaving the discussion of this area, let us consider a specific chemical example. The water molecule has C2V symmetry, hence its normal vibrational modes have A, Ai, B, or B2 symmetry. The three normal modes of H2O are pictorially depicted in Fig. 6.3.1. From these illustrations, it can be readily seen that the atomic motions of the symmetric stretching mode, iq, are symmetric with respect to C2, rv(xz) and r v(yz) thus iq has A symmetry. Similarly, it is obvious that the bending mode, i 2, also has A symmetry. Finally, the atomic motions of the asymmetric stretching mode, V3, is antisymmetric with respect to C2 and rv (xz), but symmetric with respect to r v (yz). Hence m has B2 symmetry. This example demonstrates all vibrational modes of a molecule must have the symmetry of one of the irreducible representations of the point group to which this molecule belongs. As will be shown later, molecular electronic wavefunctions may be also classified in this manner. 

Three of the representations for C2v, labeled Aj, and B2, have now been determined. The fourth, called A2, can be found by using the group properties described in Table 4.7. A complete set of irreducible representations for a point group is called the character table for that group. The character table for each point group is unique character tables for common point groups are in Appendix C. 

TABLE 4.7 Properties of Characters of Irreducible Representations in Point Groups 

The total number of symmetry operations in the group is called the order (h). To determine the order of a group, simply total the number of symmetry operations hsted in the top row of the character table. 

Symmetry operations are arranged in classes. AH operations in a class have identical characters for their transformation matrices and are grouped in the same column in character tables. 

The number of irreducible representations equals the number of classes. This means that character tables have the same number of rows and columns (they are square). 

The complete character table for C2v with the irreducible representations in the order commonly used, is 

The labels used with character tables are as follows  

The labels in the left column used to designate the representations will be described later in this section. Other useful terms are defined in Table 4-7. 

The IRs are labeled using both Bethe and Mulliken notation. 

The entry + or — signifies a positive or negative integer, respectively. 

We say that z forms a basis for A,or that z belongs to Ai, or that z transforms according to the totally symmetric representation Ai. The s orbitals have spherical symmetry and so always belong to IY This is taken to be understood and is not stated explicitly in character tables. Rx, Ry, Rz tell us how rotations about x, y, and z transform (see Section 4.6). Table 4.5 is in fact only a partial character table, which includes only the vector representations. When we allow for the existence of electron spin, the state function ip(x y z) is replaced by f(x y z)x(ms), where x(ms) describes the electron spin. There are two ways of dealing with this complication. In the first one, the introduction of a new 

Special notation is required for the complex representations of cyclic groups, and this will be explained in Section 4.7. The notation used for the IRs of the axial groups ClDOV and Dooh is different and requires some comment. The states of diatomic molecules are classified according to the magnitude of the z component of angular momentum, L , using the symbols 

All representations except S are two-dimensional. Subscripts g and u have the usual meaning, but a superscript + or is used on S representations according to whether x(oy) = l. For L 0, x(C2), an(f x(°v) are zero. In double groups the spinor representations depend on the total angular momentum quantum number and are labeled 

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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. 

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

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. 

We have seen that any two of the C2, ( Jxz), (r Jyz) elements may be regarded as generating elements. There are four possible combinations of + 1 or — 1 characters with respect to these generating elements, + 1 and + 1, + 1 and -1,-1 and +1,-1 and —1, with respect to C2 and (tJxz). These combinations are entered in columns 3 and 4 of the C2 character table in Table A.l 1 in Appendix A. The character with respect to / must always be + 1 and, just as (r Jyz) is generated from C2 and (tJxz), the character with respect to (r Jyz) is the product of characters with respect to C2 and (tJxz). Each of the four rows of characters is called an irreducible representation of the group and, for convenience, each is represented by a symmetry species Aj, A2, or B2. The A] species is said to be totally symmetric since all the characters are + 1 the other three species are non-totally symmetric. 

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 H2O molecule, therefore, has three normal vibrations, which are illustrated in Figure 4.15 in which the vectors attached to the nuclei indicate the directions and relative magnitudes of the motions. Using the C2 character table the wave functions ij/ for each can easily be assigned to symmetry species. The characters of the three vibrations under the operations C2 and (t (xz) are respectively + 1 and +1 for Vj, - - 1 and + 1 for V2, and —1 and —1 for V3. Therefore... 

Using the C2 character table (Table A. 11 in Appendix A) the characters of the vibrations under the various symmetry operations can be classified as follows ... 

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

Inspection of this character table, given in Table A. 12 in Appendix A, shows two obvious differences from a character table for any non-degenerate point group. The first is the grouping together of all elements of the same class, namely C3 and C as 2C3, and (t , and 0-" as 3o- . 

The III character table is given in Table A.46 in Appendix A. The very high symmetry of this point group results in symmetry species with degeneracies of up to five, as in and... 

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