Point symmetry elements

Derivation of selection rules for a particular molecule illustrates the complementary nature of infrared and Raman spectra and the application of group theory to the determination of molecular structure. 

The symmetry element that transforms the original equilibrium configuration into another one superimposable on the original without change in 

The selection of the axes in a coordinate system can be confusing. To avoid this, the following rules are used for the selection of the z axis of a molecule  

If a plane divides the equilibrium configuration of a molecule into two parts that are mirror images of each other, then the plane is called a symmetry 

If a molecule is rotated 360°/n about an axis and then reflected in a plane perpendicular to the axis, and if the operation produces a configuration indistinguishable from the original one, the molecule has the symmetry element of rotation-reflection, which is designated by S . 

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A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important... 

The symbol indicates a position for a possible point symmetry element. 

Table 1-4 lists the point symmetry elements and the corresponding symmetry operations. The notation used by spectroscopists and chemists, and used here, is the so-called Schoenflies system, which deals only with point groups. Crystallographers generally use the Hermann-Mauguin system, which applies to both point and space groups. 

Table 1-4 Point Symmetry Elements and Symmetry Operations...

For molecules it would seem that the point symmetry elements can combine in an unlimited way. However, only certain combinations occur. In the mathematical sense, the sets of all its symmetry elements for a molecule that adhere to the preceding postulates constitute a point group. If one considers an isolated molecule, rotation axes having n = 1,2,3,4,5,6 to oo are possible. In crystals n is limited to n = 1,2,3,4, and 6 because of the space-filling requirement. Table 1-5 lists the symmetry elements of the 32 point groups. 

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. 

Chirality, an important shape property of molecules, can be regarded as the lack of certain symmetry elements. Chirality measures are in fact measures of symmetry deficiency. These principles, originally used for crisp sets, also apply for fuzzy sets. Considering the case of three-dimensional chirality, the lacking point symmetry elements are reflection planes a and rotation-reflections 82 of even indices. Whereas the lacking symmetry elements can be of different nature in different dimensions, nevertheless, all the concepts, definitions, and procedures discussed in this section have straightforward generalizations for any finite dimension n. 

By analogy with chirality and various chirality measures, more general symmetry deficiencies and various measures for such symmetry deficiencies can be defined with reference to an arbitrary collection of point symmetry elements. We shall discuss in some detail only the three-dimensional cases of symmetry deficiencies, however, as it has been pointed out in reference [240], all the concepts, definitions, and procedures listed have straightforward generalizations for any finite dimension n. 

Evidently, if the family R contains a symmetry element of reflection o or one of the rotation-reflections S2k, then the R-sets are achiral sets. For any set T (chiral or achiral), the various extremal achiral sets can be generated by special R-sets which are extremal over all choices of families R containing at least one of the above point symmetry elements of o or S2 <. Following the notation of reference [240], subscripts a and r are used to distinguish achiral sets and R-sets. 

The unit cell has the shape of a rhomb with angles of 60° and 120° (Figure 7). The point symmetry elements (six-fold axis normal to the plane, six two-fold rotation axes in the plane, six mirrors normal to the plane, and one in the plane, inversion) pass through the unit cell origin, at the center of the hexagons. There are two symmetry-related carbon atoms in the unit cell, labeled as A and B in Figures 7 and 9, with fractional coordinates (1/3,2/3) and... 

Note that the letter is different in front and back. The other symmetry elements of the letter A are also indicated. The six basic point-symmetry elements (1, 2, 3,4, 6, and i) can describe the crystal symmetry as it is macroscopically recognizable by inspection, if needed, helped by optical microscopy. 

In this chapter, we are only concerned about the point symmetry of molecules. We use the point symmetry operations present in molecules to classify them into molecular point groups. There are five kinds of point symmetry elements that a molecule can possess, and therefore, there are also five kinds of point symmetry operations. 

TABLE 8.1 The different types of point symmetry elements and operations. 

Crystal lattices have symmetry elements such as rotation axes, mirror planes, inversion points, and combinations of these. A crystalline lattice has translation symmetry, except for quasi-crystals or icosahedral phases, which have lattices with point symmetry elements only. Glasses are amorphous solids that do not have any symmetry element in their lattices. 

This theorem has been proven using an earlier result of Pechukas along a steepest descent path the point symmetry group (as well as the framework group) of nuclear configurations may change only at a critical point, where it must have all those point symmetry elements (framework group elements, resp.) that are present at non-critical points of the path [5]. 

The vectors aj(A) have well-defined orientation with respect to point-symmetry elements of the lattices that are the same for both lattices because of the symmetrical character of the transformation (4.77). Let us define the components of the vectors aj(A) by the parameters 8 assuring their correct orientation relative to the lattice symmetry elements and the correct relations between their lengths (if there are any). Then three vector relations (4.77) give nine linear nonhomogeneous equations to determine nine matrix elements / (AA) as functions of the parameters sj,. The requirements that these matrix elements must be integers define the possible values of the parameters Sk giving the solution of the problem. 

The S2N2 reaction also provides a goo4 example of the occurrence and consequences of this type of non-uniqueness. Although both dimer and final polymer phases have the same space group (P2-./G) and the same number of SN groups per unit cell, the z/m point symmetry elements of the two phases are rotated with respect to each other by 90°. As a consequence, the reaction product (SN)x bas the overall... 

It is the mentioned symmetry properties additional to the discrete translational symmetry that lead to a classification of the various possible point lattices by five Bravais lattices. Like the translations, these symmetry operations transform the lattice into itself They are rigid transforms, that is, the spacings between lattice points and the angles between lattice vectors are preserved. On the one hand, there are rotational axes normal to the lattice plane, whereby a twofold rotational axis is equivalent to inversion symmetry with respect to the lattice point through which the axis runs. On the other hand, there are the mirror lines (or reflection lines), which he within the lattice plane (for the three-dimensionaUy extended surface these hnes define mirror planes vertical to the surface). Both the rotational and mirror symmetry elements are point symmetry elements, as by their operation at... 

The Bravais lattices are classified according to the applying group of rotational and mirror symmetries. Figure 4.11 presents these classes with the translation vectors and the point symmetry elements indicated within one unit cell. 

It is useful to group the point symmetry elements passing through any point within a unit cell. First of all, there are the five groups of rotational symmetry including... 

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