Symmetry element restrictions

Symmetry element restrictions on the number of allowed q values in the expansion of the crystal-field potential... 

The interpretation of chemical reactivity in terms of molecular orbital symmetry. The central principle is that orbital symmetry is conserved in concerted reactions. An orbital must retain a certain symmetry element (for example, a reflection plane) during the course of a molecular reorganization in concerted reactions. It should be emphasized that orbital-symmetry rules (also referred to as Woodward-Hoffmann rules) apply only to concerted reactions. The rules are very useful in characterizing which types of reactions are likely to occur under thermal or photochemical conditions. Examples of reactions governed by orbital symmetry restrictions include cycloaddition reactions and pericyclic reactions. 

Mathematical symmetry is a little more restrictive than is the meaning of the word in everyday usage. For example, some might say tliat flowers, diamonds, butterflies, snail shells, and paisley ties (Fig. 3.1) are all Itighly symmetrical because of (lie harmony and attractiveness of their forms and proportions, but the pattern of a paisley tie is not balanced in mathematical language, it lacks symmetry elements. A (lower, crystal, or molecule is said to liave symmetry if it has two or more orientations in space that are indistinguishable, and the criteria forjudging these are based on symmetry elements and symmetry operations. 

Ostensibly, only allowed transitions should be observed experimentally. In many cases, however, transitions are observed which formally are forbidden. This is not as disastrous as it would appear. Usually it is our model of the molecular structure which is wrong we assume a static molecular skeleton and forget that vibrations can change this firm geometry and allow the molecule to have other structures. These other structures have different symmetry elements from those we worked with and give new and different selection rules. For example, we could destroy the inversion center and remove the parity restriction. 

Having established that the mirror plane is a proper symmetry element even if the chains are distorted, we look at HOMO-LUMO interactions. If we restrict our attention to chains with even numbers of electrons and focus on the HOMO and the LUMO, we can see that there are only two kinds of chains those in which the HOMO is symmetric and the LUMO antisymmetric, and those in which the HOMO is antisymmetric and the LUMO symmetric. Goldstein and Hoffmann have named these types respectively Mode 2 and Mode 0.14 Table 10.1 shows a few examples. Note that anions and cations are covered as well as neutral molecules. 

A fuzzy set B is called an R-deficient set if B has none of the point symmetry elements of family R. However, by analogy with the case of crisp sets, it takes only infinitesimal distortions to lose a given symmetry element. Consequently, unless further restrictions are applied, the total mass difference between a fuzzy set of a specified symmetry and another fuzzy set that does not have this symmetry can be infinitesimal. As a result, i -deficient fuzzy sets and fuzzy R sets can be almost identical. Nevertheless, the actual symmetry deficiencies of fuzzy continua, such as formal molecular bodies represented by fuzzy clouds of electron densities, can be defined in terms of the deviations from their maximal R subsets and minimal R supersets, defined in subsequent text. 

Although the word crystal in its everyday usage is almost synonymous with symmetry, there are severe restrictions on crystal symmetry. While there are no restrictions in principle on the number of symmetry classes of molecules, this is not so for crystals. AH crystals, as regards their form, belong to one or another of only 32 symmetry classes. They are also called the 32 crystal point groups. Figures 9-12 and 9-13 illustrate them by examples of actual minerals and by stereographic projections with symmetry elements, respectively. 

To have 32 symmetry classes for the external forms of crystals is a definite restriction, and it is obviously the consequence of inner structure. The translation periodicity limits the symmetry elements that may be present in a crystal. The most striking limitation is the absence of fivefold rotation in the world of crystals. Consider, for example, planar networks of regular polygons (Figure 9-16). Those with threefold, fourfold, and sixfold symmetry cover the... 

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