Linear species symmetry properties

Our concern is with the electronic structure of crystals and it is our hope that it will prove possible to describe the symmetry properties of the crystal orbitals of a crystal, for example, by something akin to the irreducible representations used to describe the molecular orbitals of molecules. By analogy with the molecular case it would be expected that the symmetry species of the crystal orbital which contains no nodes would be important (in the molecular case all spectroscopic selection rules are related to it, for instance). This will be the crystal counterpart of a point group totally symmetric irreducible representation (which contains no symmetry-required nodes). For the case of the crystal, represent this zero-node symmetry pattern by a dot on a piece of paper. Now do the same for corresponding parallel one-node, two-node, three-node and so on patterns (a direction will have to be chosen to which the nodes are all perpendicular but this apparent arbitrariness will pose no problem). Clearly, the dots should not be drawn randomly scattered around but, rather, related to one another in a sensible way. In Fig. 17.4 is shown how it can be done for the linear polyene of Fig. 17.2. The task is now to do the same for the crystal of Fig. 17.3. A difference is that each dot will now carry two indices, indicating the number of nodes in each of the two directions in the two-dimensional crystal of Fig. 17.3. This is easy when the nodes are perpendicular to either the x or y axes. 

A. symmetry coordinate of. species constructed from the bond stretches is also reiiuired. FAudently one cannot simply operate upon ri, since n is symmetric with respect to E, C3, C, linear combination, namely, ri + fi, has the desired property it is sent into itself by E, C2, cn, and 0-34, just as is an-Therefore... 

---