Matrices character tables

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. 

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. 

Because the trace of a matrix is independent of the coordinate system, matrices representing operations that have the same effect in different coordinate systems must have the same trace. It is possible to use this fact to abbreviate the character tables. For example, consider the long and short versions of the character table of ... 

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. 

It is relatively easy to deduce the four one-dimensional representations. As in every group, there must be the so-called totally symmetric representation, in which every symmetry operation is represented by the one-dimensional matrix 1. At this point, we have in hand the following part of the character table ... 

It is easy to see that the operation C2 transforms into the negative of itself and v36 into itself. Thus a matrix is obtained which has only the diagonal elements -1 and 1 and the character 0 as required by the character table. It is equally easy to see that oh carries each component of i 3 into itself, so that the matrix of the transformation has only the diagonal elements I and 1 and hence a character of 2. We could carry out similar reasoning for the remaining operation applied to v, and v36 and also with respect to the application of all of the operations in the group to v and vAh, and it would be found that they satisfy the requirements of the characters of the E representation in every respect. 

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... 

We list here full matrix representations for several groups. Abelian groups are omitted, as their irreps are one-dimensional and hence all the necessary information is contained in the character table. We give C3v (isomorphic with D3) and C4u (isomorphic with D4 and D2d). By employing higher 1 value spherical harmonics as basis functions it is straightforward to extend these to Cnv for any n, even or odd. We note that the even n Cnv case has four nondegenerate irreps while the odd n Cnv case has only two. 

To find the symmetry of the normal modes we study the transformation of the atomic displacements xL y, z,, i 0,1,2,3, by setting up a local basis set e,i ci2 el3 on each of the four atoms. A sufficient number of these basis vectors are shown in Figure 9.1. The point group of this molecule is D3h and the character table for D3h is in Appendix A3. In Table 9.1 we give the classes of D3h a particular member R of each class the number of atoms NR that are invariant under any symmetry operator in that class the 3x3 sub-matrix r, (R) for the basis (e,i el2 cl31 (which is a 3 x 3 block of the complete matrix representative for the basis (eoi. .. e331) the characters for the representation T, and the characters for... 

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. 

It was discussed before that the irreducible representations can be produced from the reducible representations by suitable similarity transformations. Another important point is that the character of a matrix is not changed by any similarity transformation. From this it follows that the sum of the characters of the irreducible representations is equal to the character of the original reducible representation from which they are obtained. We have seen that for each symmetry operation the matrices of the irreducible representations stand along the diagonal of the matrix of the reducible representation, and the character is just the sum of the diagonal elements. When reducing a representation, the simplest way is to look for the combination of the irreducible representations of that group—that is, the sum of their characters in each class of the character table—that will produce the characters of the reducible representation. 

The set of Cartesian displacement vectors as basis for a representation is shown in Figure 5-9. The symmetry operations of the point group are also shown. The D h character table is given in Table 5-3. Recall (Chapter 4) that the matrix of rotation by an angle is... 

In fact the relative coefficients within a set of symmetrically equivalent atoms, such as Cl, C3, C5, and Cg in para-benzosemiquinone, can be determined by group theory alone. The appropriate set of symmetrized orbitals, also listed in Table 1, can be obtained by use of character tables and procedures described in Refs. 3 to 6. In matrix notation, the symmetrized combinations are <1> = Up, where <1) and p are column vectors and U is the transformation matrix giving the relations shown in part (c) of Table 1. The transformation U and its transpose U can be used to simplify the solution of the secular matrix X since the matrix multiplication UXU gives the block diagonal form shown in part (6) of Table 1. 

---