D2h, character table

The symmetry operations of the molecule place it in the point group D2h. The D2h character table is given in Table 7-1. 

The rules for the state correlation diagrams are the same as for the orbital correlation diagrams only states that possess the same symmetry can be connected. In order to determine the symmetries of the states, first the symmetries of the MOs must be determined. These are given for the face-to-face dimerization of ethylene in Table 7-1. The D2h character table (Table 7-2) shows that the two crucial symmetry elements are the symmetry planes a(xy) and v"(yz). The MOs are all symmetric with respect to the third plane, vide supra). The corresponding three symmetry operations will unambiguously determine the symmetry of the MOs. Another possibility is to take the simplest subgroup of D2t, which already contains the two crucial symmetry operations, that is, the C2v point group (cf.,... 

Using the D2h character table shown, verify that the group orbitals in Figure 5.18 match their irreducible representations. 

The normal modes of vibration of C2H4 are shown below. By inspection of the D2h character table, deduce the symmetry species to which the normal modes belong. The coordinate system adopted is shown on the right. 

Let us now apply this method to a specific example. Consider the ethylene molecule with D2h symmetry. As can be seen from the character table of the L 2h point group (Table 6.4.2), this group has eight symmetry species. Hence the molecular orbitals of ethylene must have the symmetry of one of these eight representations. In fact, the ground electronic configuration for ethylene is... 

The character table of a point group defines the symmetry properties of a (wave) function as either 1 for symmetric or —1 for antisymmetric with respect to each symmetry operation/ The first row lists all the symmetry operations of the point group and the first column lists the Mulliken symbols of all possible irreducible representations, the symmetry transformation properties that are allowed for wavefunctions. As an example, the character table for the D2h point group is given in Table 4.1. The character tables of all relevant point groups are given in many textbooks.134,273-275 The last column shows the transformation properties of the axes x, y and z, which are used to determine electronic dipole and transition moments (Section 4.5). 

Table 2.3 Character table for the irreducible representations of group D2h-...

In this D2h case, unlike in the C- y point group, each operation is in its own class, and the number of columns above is identical to that in the >2 character table. SALC(Ag) and SALC(5i ) of these oxygen 2s wave functions can be obtained by multiplication of each outcome by the characters associated with each operation of these irreducible representations, followed by addition of the results ... 

Every book on group theory contains the table of characters of the symmetry group D2h (see Table C.6 the x-axis perpendicular to the plane of the molecule, and z goes through the nitrogen atoms). 

As illustrated in Fig. 1.2 poly(po/p-phenylene), possesses planes of symmetry through both the major and minor axes. This is denoted as D2h symmetry. The character table for D2h symmetry is shown in Table 11.3. 

Table 11.3 The experimentally determined and theoretical predictions of the low-lying vertical excitations of biphenyl (in eV) (The experimental assignments are from (Bursill et al. 1998). Pariser-Parr-Pople calculations with tp = 2.539 eV, te = 2.22 eV, and U = 10.06 eV (Bursill et al. 1998). This table also serves to define the character table for D2h symmetry group.)...

In the Character Table of D2h we find four representations that are symmetric under (twofold) rotation about the z axis, and can thus be related to... 

The reader who has not skipped Section 2.3 is advised to look at Table 3.1 in conjuction with the Character Table of Dqo/i in Appendix A. Several additional points will then emerge a) Each of its two-dimensional irreps splits to two of D2h, b) there is an infinite number of two-dimensional representations, indicated in the table by each of which splits similarly c) D2/, is the co-kernel of Ag (not its kernel, which is C,), as can be seen from the fact that Ag maps onto the totally symmetric representation of D2/1, indicating that the quadrupolar field can be regarded as a perturbation which has the symmetry species Ag in Dooh ... 

Information on the irreducible representations of the various point groups is presented customarily in tables that are called character tables because they give the character of the irreducible representatimi of each symmetry operation in a point group. The character of a representation is the trace (the sum of the diagonal elements) of the matrix that represents that operatimi. Character tables for the D2h, C2v and 4 point groups are presented in Tables 4.2, 4.3 and 4.4. The symmetry elements of the point group are displayed in the top row of each table, and the conventional names, called Mulliken symbols, of the irreducible representations are given in the first column. Letters A and B are used as the Mulliken symbols for... 

At this point we consult Table 2.2 and see that is the only irrep in which —1 is the character of each of the four symmetry operations in the eecond, excluded set. Conversely, only Bzu along with the totally symmetric representation - has 1 as the character of each of the sym-ops retained in Cf. This latter fact is expressed by the statement that the subgroup is the ker nel of the irrep B u of the parent group D2h- We note further that B u is the representation of the coordinate x, and realize that, after the system has been perturbed by a dipolar field along the x axis, the energy of an electron in a px orbital can remain unaffected only under sym-ops that do not convert x to —x. The perturbation, which has the representation Bs has evidently reduced the symmetry of the system to that of its kernel, i.e. from D2h fo its subgroup... 

Table 7.11 The reducible representation for the valence orbitals of the carbon atoms of ethene (D2h). In the application of the reduction formula (lower table) only the nonzero characters from r(C(2s, 2p)) need be considered.

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