Z>2h point group

Usang the method outiined in Section 4.4, confirm that AII3 and Al2l belong to the and Z>2h point groups, respectively. 

So the eight pairs of electrons of this molecule occupy delocalized molecular orbitals lag to 1 3U, while the first vacant orbital is l g- Note that the names of these orbitals are simply the symmetry species of theZ)2h point group. In other words, molecular orbitals are labeled by the irreducible representations of the point group to which the molecule belongs. So for ethylene there are three filled orbitals with Ag symmetry the one with the lowest energy is called lag, the next one is 2ag, etc. Similarly, there are two orbitals with Z iu symmetry and they are called lb u and 2bi . All the molecular orbitals listed above, except the first two, are illustrated pictorially in Fig. 6.4.2. By checking the >2h character table with reference to the chosen coordinate system shown in Fig. 6.4.2, it can be readily confirmed that these orbitals do have the labeled symmetry. In passing, it is noted that the two filled molecular orbitals of ethylene not displayed in Fig. 6.4.2, lag and l iu, are simply the sum and difference, respectively, of the two carbon Is orbitals. 

Groups with one main axis of symmetry the z axis points along the main axis of symmetry and, whore applicable, the x axis lies in one of the rrT planes or coincides with one of the C2 axes. For 2 and 2h the x, y, z axes coincide with the three equivalent two-fold axes. 

This group has symmetry element E, a principal Cn axis, n secondary C2 axes perpendicular to Cn, and a ah also perpendicular to C . The necessary consequences of such combination of elements are a S axis coincident with the Cn axis and a set of n ctv s containing the C2 axes. Also, when n is even, symmetry center i is necessarily present. The BrF molecule has point group symmetry D4h, as shown in Fig. 6.1.8. Examples of other molecules belonging to point groups >2h, D3h, Z>5h and D6h are given in Fig. 6.2.6. 

Some molecular orbitals of ethylene (point group Z>2h)- All are completely filled except l 2g in the ground state. The x axis points into the paper. 

Perhaps the most important feature of character tables for our purposes is that we can determine the symmetry of a product of two irreducible representations simply by looking at the products of the characters of these representations in the same point group. For example, the product of the HOMO and LUMO of ethylene transforms as B2 in E>2h, as you can see by comparing the products of the characters of Bj,g and with the corresponding characters of B2 - Alternatively, we can just note that the product of B g and Ri transforms as the product of z and yz, or yz, which... 

A more complex example may be represented by TaSe2 its modification called 2H-TaSe2 is hexagonal (space group P6 mmc, with two formula units in the unit cell). This layered compound shows a displacive 2D modulation (defined by two vectors) its symmetry may be therefore described in terms of a supergroup in a 5D superspace (Janner and Janssen 1980). A general point is therefore denoted by the 5 parameters x,y,z, t, u, and a position vector by the five components xa + yb + zc + td + ue of the superspace, with a, b, c basis in the position space and d and e in the internal subspace . 

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