Point Groups and Character Tables

The symmetry elements of the water molecule are easily detected. There is only one proper axis of symmetry, which is the one that bisects the bond angle and contains the oxygen atom. It is a C2 axis and the associated operation of rotating the molecule about the axis by 180° results in the hydrogen atoms exchanging places with each other. The demonstration of the effectiveness of the operation is sufficient for the diagnosis of the presence of the element. 

The four symmetry elements form a group, which may be demonstrated by introducing the appropriate rules. The rules are exemplified by considering the orbitals of the atoms present in the molecule. Such a consideration also develops the relevance of group theory, in that it leads 

The electrons which are important for the bonding in the water molecule are those in the valence shell of the oxygen atom 2s22p4. It is essential to explore the character of the 2s and 2p orbitals, and this is done by deciding how each orbital transforms with respect to the operations associated with each of the symmetry elements possessed by the water molecule. 

The character of an orbital is symbolized by a number which expresses the result of any particular operation on its wave function. In the case of the 2s orbital of the oxygen atom which is spherically symmetrical, there is no change of sign of j/ with any of the four operations E, C2 (z), 7V (xz) and ov (yz). These results may be written down in the form  

The 2py orbital of the oxygen atom changes sign when the C2 operation is applied to it, and when it is reflected through the xz plane, but is symmetric with respect to the molecular plane, yz. The representation expressing the character of the 2py orbital is another member of the group  

---

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). 

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. 

This appendix briefly summarizes molecular symmetry (symmetry elements, point groups and character tables) of a molecule so as to facilitate readers understanding of the molecular structures and molecular orbitals presented in this chapter. We use some speciflc examples such as NH3 with Csv symmetry, CH3 radical with >3 symmetry (see Fig. 5.18) and c-C4Fg radical with D4h symmetry (Fig. 5.3) to introduce and illustrate some important aspects of the molecular symmetry. 

Acetylene (HC=CH) belongs to the point group whose character table is given in Table A.37 in Appendix A, and its vibrations are illustrated in Figure 6.20. Since V3 is a vibration and T T ) = 2"+, the 3q transition is allowed and the transition moment is polarized along the z axis. Similarly, since Vj is a vibration, the 5q transition is allowed with the transition moment in the xy plane. 

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. 

As an example, consider IR activity of the six normal vibrations of the NH3 molecule, which are classified into 2A and 2E species of C3V point group. The character table shows that fiz belongs to the A and the pair of (px, fiy) belongs to the E species. Thus, all six normal vibrations are IR-active. 

The numbers in the table, the characters, detail the effect of the symmetry operation at the top of the colurrm on each representation labelled at the front of the row. The mirror plane that contains the H2O molecule, a (xz), leaves an orbital of bi symmetry unchanged while a Ci operation on the same basis changes the sign of the wavefimction (orbital representations are always written in the lower case). An orbital is said to span an irreducible representation when its response upon operation by each symmetry element reproduces the same characters in the row for that irreducible representation. For atoms that fall on the central point of the point group, the character table lists the atomic orbital subscripts (e.g. x, y, z as p , Pj, p ) at the end of the row of the irreducible representation that the orbital spans. A central s orbital always spans the totally synunetric representation (aU characters = 1). For the central oxygen atom in H2O, the 2s orbital spans ai and the 2px, 2py, and 2p span the bi, b2, and ai representations, respectively (see (25)). If two or more atoms are synunetry equivalent such as the H atoms in H2O, the orbitals must be combined to form symmetry adapted hnear combinations (SALCs) before mixing with fimctions from other atoms. A handy mathematical tool, the projection operator, derives the functions that form the SALCs for the hydrogen atoms. 

Appendix 3 gives character tables for the most commonly encountered point groups, and each table has the same format as those in Tables 3.2 and 3.3. 

Table 6.18. Character table for theD4(, point group and characters of the reducible representation Fa of a square-planar complex ML4...

Formaldehyde has symmetry operations which place it in the point group. The character table for was given in Table 6-1. Since 0(py) transforms as bj and OCp ) as b, the ground... 

Figure 3.23 shows the two point groups that arise for linear molecules. If the two ends of the molecule are different, then the only symmetry elements are and the vertical mirror planes, so the point group is Coov by analogy with C2v, Csv, etc. If the molecule has equivalent points at either end of the axis then it will also have a horizontal mirror plane

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. 

This result is similar to that for e x e, in Equation (4.29), in the point group and can be verified using the Tig , character table in Table A.36 in Appendix A. As the two electrons (or one electron and one vacancy) in the partially occupied orbitals may have parallel (S = 0) or antiparallel (5=1) spins there are six states arising from the configuration in Equation... 

Follow the procedure used in the text in obtaining the character table for the C2 point group and develop the character table for the C3 point group. 

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 ... 

To find the irreducible representations which can be produced from the orbitals we need the characters " (l ) and the effect of Om on the orbitals. Taking the last point first, we find 0Mg for all R of B and g = Pi Pa Pa dt, da da, d4, d6 and the results are given in Table 7-9,2, We have carried out this kind of step before (see 5-9) and for this particular point group and axis choice, the process is particularly simple, for example we have... 

Table 12.3. Multiplication table, factor tables, and character tables for the point group C,.

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. 

The factor groups are isomorphic (one-to-one correspondence) with the 32 point groups and, consequently, the character table of the factor group can be obtained from the corresponding isomorphic point group. 

Next, we discuss representations and character tables. A group of order h can be represented by h matrices, each of dimensions hxh. Flowever, there exist so-called irreducible representations for each group, which are block-diagonal submatrices, which span the space. Table 7.6 presents the character tables for all 32 crystallographic point groups, and some other groups as well. Table 7.7 is an abbreviated form of Table 7.6. 

---