Character tables notation

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. 

Another common notation for the irreducible representations is the so-called Bethe notation, in which the representations are denoted by E, symbols (i = 1,2,...), where the subscript i denotes the dimension. It is not simple to establish an equivalence between these two types of notation, since it depends on the symmetry group. For the moment, we will just mention both notations so that readers will be famihar with any character table. 

Linear molecules belong to either 6 or the character tables are given in Section 9.12. The operation Ca(z) is a rotation by an angle a about the CM axis. In addition to the Mulliken notation, the spectroscopic symbols 2,n,A,... are also given. As we saw in Section 1.19, apart from spin degeneracy, 2 electronic levels are nondegenerate (A), whereas... 

A practical consequence of collecting all operations in the same class when writing down the complete set, for example, at the head of a character table, is that the notation used is a little different from what we have beep using thus far. This new and final form of notation will now be explained and illustrated for the four kinds of symmetry operations. 

Symmetry Notation.—A state is described in terms of the behavior of the electronic wave function under the symmetry operations of the point group to which the molecule belongs. The characters of the one-electron orbitals are determined by inspection of the character table the product of the characters of the singly occupied orbitals gives the character of the molecular wave function. A superscript is added on the left side of the principal symbol to show the multiplicity of the state. Where appropriate, the subscript letters g (gerade) and u (ungerade) are added to the symbol to show whether or not the molecular wave function is symmetric with respect to inversion through a center of symmetry. 

The character table of Oh is in Appendix A3. The BSW notation for the IRs depends on compatibility relations which are derived in Table 17.5. 

Exercise 17.4-5 The E point lies on [1 1 0] between T and M in the kz = 0 plane (Figure 16.12(b)). What is P(k) at E List the basis functions for the IRs, naming them in both Mulliken and BSW notation. Note that E, A, and T all lie in the (1 10) plane through T defined by x y 0. Can the states A2, T, and T2 exist in the same energy band as a E2 state What other E state is compatible with these A and T states [Hint These basis functions will differ from those usually seen in character tables with vertical planes x = 0, y 0 here the vertical planes arc z 0, x y 0.]... 

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. 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

When operations such as C3 are in the same class, the listing in a character table is 2C3, indicating that the results are the same whether rotation is in a clockwise or counterclockwise direction (or, alternately, that C3 and give the same result). In either case, this is equivalent to two columns in the table being shown as one. Similar notation is used for multiple reflections. 

Table B.4. Double group characters table for the Ta point group. The numbers before the symmetry operations correspond to the number of geometrically different axes or symmetry planes. Some of the operations of the double group belong to the same class as those of the original group. When more than one IR is indicated, the first one corresponds to the notation of Mulliken [11], the second one to Koster et al. [9] and the one in parentheses to [3]...

Table 3.3 shows the character table for the Cgv point group. The NH3 molecule possesses C3V symmetry, and worked example 3.2 illustrated the principal axis of rotation and planes of symmetry in NH3. In the character table, the presence of three cr planes in NH3 is represented by the notation 3ctv in the top line of the table. The notation 2C3 summarizes the two operations C3 and C3. The operation C3 is equivalent to the identity operator, E, and so is not specihed again. 

For molecules with other symmetries there are other character tables and a similar notation for the symmetry species. The wave functions of asymmetric molecules cannot be classified into symmetry species. 

Now compare this row of characters with the rows in the C2V character table. There is a match with the row for symmetry type Ai, and therefore the symmetric stretch is given the Ai symmetry label. Now consider the asymmetric stretching mode of the SO2 molecule. The vectors (Figure 4.12b) are unchanged by the E and cr (yz) operations, but their directions are altered by rotation about the C2 axis and by reflection through the a (xz) plane. Using the notation that a T means no change , and a -1 means... 

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