Group theory character tables

An introduction to the mathematics of group theory for the non-mathematician. If you want to learn formal group theory but are uncomfortable with much of the mathematical literature, this book deserves your consideration. It does not treat matrix representations of groups or character tables in any significant detail, however. 

Elliot, R. J. (1954) Spin-orbit coupling in band theory - character tables for some double space groups. Phys. Rev. 96, 280-7. 

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. 

In practice, it is the characters of irreducihle representations that are used in most chemical applications of group theory. This means that one needs only the character table for a group, rather than the whole representation table. For the Csv group, the character table is displayed in Table 13-18. Comparison with Table 13-13 will make clear that the character is merely the sum of diagonal elements. (For one-dimensional representations, the character and the representation are identical.) Since the representation for the identity operation is always a unit matrix, the character for this operation is always the same as the dimension of the representation. Hence, the first character in a row tells us the dimension of the corresponding representation. 

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. 

This result (problem 14) allows the coefficients to be calculated. Thus, with a knowledge of the symmetry group and the corresponding table of characters, the structure of the reduced representation can be determined. Equation (37) is of such widespread applicability that it is referred to by many students of group theory as the magic formula. ... 

It should now be apparent how the species A1( A2, Bv and B2 arise. Character tables have been worked out and are tabulated for all the common point groups. Presenting all the tables here would go beyond the scope of the discussion of symmetry and group theory as used in this book. However, tables for some common point groups are shown in Appendix B. 

Thus, any representation Tcan be expressed as a function of its irreducible representations Pi. This operation is written as P = S a, Pi, where a, indicates the number of times that Pi appears in the reduction. In group theory, it is said that the reducible representation P is reduced into its Pi irreducible representations. The reduction operation is the key point for applying group theory in spectroscopy. To perform a reduction, we need to use the so-called character tables. 

GROUP THEORY AND SPECTROSCOPY Table 7.6 The character table of group D3... 

Chapter 7 is a very simple introduction to group theory and its usefulness to interpreting the optical spectra of active centers. The purpose of this chapter is to present some basic concepts, for non-specialists in group theory, so they can evaluate its potential and, hopefully the feasibility of applying it to simple problems, such as the determination and labeling of the energy levels of an active center by means of the character table of its symmetry group. 

The symbols used for the representations are those proposed by Mulliken. The A representations are those which are symmetric with respect to the C2 operation, and the Bs are antisymmetric to that operation. The subscript 1 indicates that a representation is symmetric with respect to the ov operation, the subscript 2 indicating antisymmetry to it. No other indications are required, since the characters in the o column are decided by another rule of group theory. This rule is the product of any two columns of a character table must also be a column in that table. It may be seen that the product of the C2 characters and those of gv give the contents of the The representations deduced above must be described as irreducible representations This is because they... 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

Pi orbitals, 68, 310, 317 Pi-star orbital, 309, 310 Planarity, 224-225 Planck, M., 122 Plane of symmetry, 53 Plane-polarized wave, 114 Plane polar normal coordinates, 272 Point groups, 53-56,388-389 character tables of, 458-462 see also Group theory... 

Character tables are what most quantum chemists remember best of their group theory, but it is convenient to complete this review of the fundamentals by examining first the character table for a group and then the full matrix irreps for that group. As an iluustration, we shall examine the group C3 (or the isomorphic D3). Here is the character table in the same format as used in the Tables provided for this course. The... 

In otder to construct such sets of orbitals, it is most convenient to make use of group theory. Each set of equivalent directed valence orbitals has a characteristic symmetry group. If the operations of this group are performed on the orbitals, a representation, which is usually reducible, is generated. By means of the character table of the group7 this representation, which we shall call the a representation, can be reduced to its component irreducible representations. The s, p, and d orbitals of the atom also form representations of the group, and can also be divided into sets which form irreducible representations.8... 

This appendix contains tables of characters for vector and spinor representations of the point groups G that are encountered most commonly in practical applications of group theory in chemical physics. Correlation tables are given separately in Appendix A4. 

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. 

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. 

---