Element of symmetry

These simple examples serve to show that instinctive ideas about symmetry are not going to get us very far. We must put symmetry classification on a much firmer footing if it is to be useful. In order to do this we need to define only five types of elements of symmetry - and one of these is almost trivial. In discussing these we refer only to the free molecule, realized in the gas phase at low pressure, and not, for example, to crystals which have additional elements of symmetry relating the positions of different molecules within the unit cell. We shall use, therefore, the Schdnflies notation rather than the Hermann-Mauguin notation favoured in crystallography. 

In discussing molecular symmetry it is essential that the molecular shape is accurately known, commonly by spectroscopic methods or by X-ray, electron or neutron diffraction. 

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One example of a quantitative measure of molecular chirality is the continuous chirality measure (CCM) [39, 40]. It was developed in the broader context of continuous symmetry measures. A chital object can be defined as an object that lacks improper elements of symmetry (mirror plane, center of inversion, or improper rotation axes). The farther it is from a situation in which it would have an improper element of symmetry, the higher its continuous chirality measure. 

Continuous chirality measure is then defined as follows given a configuration of points P = I, its chirality content is determined by finding the nearest configuration of points Pi - 2 which has an improper element of symmetry, and by calculating the distance between the two sets using Eq. (26). 

A molecular model is a great help in visualizing not only these particular axes but all elements of symmetry. 

All molecules possess the identity element of symmetry, for which the symbol is / (some authors use E, but this may cause confusion with the E symmetry species see Section 4.3.2). The symmetry operation / consists of doing nothing to the molecule, so that it may seem too trivial to be of importance but it is a necessary element required by the mles of group theory. Since the C operation is a rotation by 2n radians, Ci = I and the symbol is not used. 

In the early days following the discovery of chirality it was thought that only molecules of the type CWXYZ, multiply substituted methanes, were important in this respect and it was said that a molecule with an asymmetric carbon atom forms enantiomers. Nowadays, this definition is totally inadequate, for two reasons. The first is that the existence of enantiomers is not confined to molecules with a central carbon atom (it is not even confined to organic molecules), and the second is that, knowing what we do about the various possible elements of symmetry, the phrase asymmetric carbon atom has no real meaning. 

We have seen in Section 4.1.4 that = n and that S2 = i, so we can immediately exclude from chirality any molecule having a plane of symmetry or a centre of inversion. The condition that a chiral molecule may not have a plane of symmetry or a centre of inversion is sufficient in nearly all cases to decide whether a molecule is chiral. We have to go to a rather unusual molecule, such as the tetrafluorospiropentane, shown in Figure 4.8, to find a case where there is no a or i element of symmetry but there is a higher-fold S element. In this molecule the two three-membered carbon rings are mutually perpendicular, and the pairs of fluorine atoms on each end of the molecule are trans to each other. There is an 54 axis, as shown in Figure 4.8, but no a or i element, and therefore the molecule is not chiral. 

In the case of CFIFClBr, shown in Figure 4.7, there is no element of symmetry at all, except the identity /, and the molecule must be chiral. 

A property such as a vibrational wave function of, say, H2O may or may not preserve an element of symmetry. If it preserves the element, carrying out the corresponding symmetry operation, for example (t , has no effect on the wave function, which we write as... 

If atoms, molecules, or ions of a unit cell are treated as points, the lattice stmcture of the entire crystal can be shown to be a multiplication ia three dimensions of the unit cell. Only 14 possible lattices (called Bravais lattices) can be drawn in three dimensions. These can be classified into seven groups based on their elements of symmetry. Moreover, examination of the elements of symmetry (about a point, a line, or a plane) for a crystal shows that there are 32 different combinations (classes) that can be grouped into seven systems. The correspondence of these seven systems to the seven lattice groups is shown in Table 1. 

By applying these rules and recognizing the elements of symmetry present in the molecule, it is possible to construct MO diagrams for more complex molecules. In the succeeding paragraphs, the MO diagrams of methane and ethylene are constructed on the basis of these kinds of considerations. 

This geometry possesses three important elements of symmetry, the molecular plane and two planes that bisect the molecule. All MOs must be either symmetric or antisymmetric with respect to each of these symmetry planes. With the axes defined as in the diagram above, the orbitals arising from carbon 2p have a node in the molecular plane. These are the familiar n and n orbitals. 

Compounds in which one or more carbon atoms have four nonidentical substituents are the largest class of chiral molecules. Carbon atoms with four nonidentical ligands are referred to as asymmetric carbon atoms because the molecular environment at such a carbon atom possesses no element of symmetry. Asymmetric carbons are a specific example of a stereogenic center. A stereogenic center is any structural feature that gives rise to chirality in a molecule. 2-Butanol is an example of a chiral molecule and exists as two nonsuperimposable mirror images. Carbon-2 is a stereogenic center. 

Since the presence of a plane of symmetry in a molecule ensures that it will be achiral, one a q)ro h to classification of stereoisomers as chiral or achiral is to examine the molecule for symmetry elements. There are other elements of symmetry in addition to planes of symmetry that ensure that a molecule will be superimposable on its mirror image. The trans,cis,cis and tmns,trans,cis stereoisomers of l,3-dibromo-rranj-2,4-dimethylcyclobutaijte are illustrative. This molecule does not possess a plane of symmetry, but the mirror images are superimposable, as illustrated below. This molecule possesses a center of symmetry. A center of symmetry is a point from which any line drawn through the molecule encouniters an identical environment in either direction fiom the center of ixnimetry. 

Indicate which of the following molecules are chiral and which are achiral. For each molecule that is achiral, indicate the element of symmetry that is present in the molecule. 

Figure 11.3 illustrates the classification of the MOs of butadiene and cyclobutene. There are two elements of symmetry that are common to both s-cw-butadiene and cyclobutene. These are a plane of symmetry and a twofold axis of rotation. The plane of symmetry is maintained during a disrotatory transformation of butadiene to cyclobutene. In the conrotatory transformation, the axis of rotation is maintained throughout the process. Therefore, to analyze the disrotatory process, the orbitals must be classified with respect to the plane of symmetry, and to analyze the conrotatory process, they must be classified with respect to the axis of rotation. 

Many (but not all) of the geometrie shapes that appear in the erystalline state are readily reeognized as being to some degree symmetrieal, and this faet ean be used as a means of erystal elassifieation. The three simple elements of symmetry that ean be eonsidered are... 

The eube for example, has 1 eentre of symmetry, 13 axes of symmetry and 9 planes of symmetry (Figure 1.1). An oetahedron also has 23 elements of symmetry and is therefore erystallographieally related to the eube. 

There are only 32 possible eombinations of the above-mentioned elements of symmetry (ineluding the asymmetrie state), and these are ealled the 32 elasses or point groups. For eonvenienee these 32 elasses are grouped into 7 systems eharaeterized by the angles between their x, y and z axes. Crystals of eourse, ean exhibit eombination forms of the erystal systems. 

An object has symmetry when certain parts of it can be interchanged with others without altering either the identity or the apparent orientation of the object. For a discrete object such as a molecule 5 elements of symmetry can be envisaged ... 

These elements of symmetry are best recognized by performing various symmetry operations, which are geometrically defined ways of exchanging equivalent parts of a molecule. The 5 symmetry operations are ... 

The complete set of symmetry operations that can be performed on a molecule is called the symmetry group or point group of the molecule and the order of the point group is the number of symmetry operations it contains. Table A2.1 lists the various point groups, together with their elements of symmetry and with examples of each. 

It is instructive to add to these examples from the numerous instances of point group symmetry mentioned throughout the text. In this way a facility will gradually be acquired in discerning the various elements of symmetry present in a molecule. 

Although carpanone s complex structure possesses no element of symmetry, it was suggested1 that carpanone could form in nature through an intramolecular cycloaddition of a C2-symmetric bis(qui-... 

The molecules shown in Fig. 1 are planar thus, the paper on which they are drawn is an element of symmetry and the reflection of all points through the plane yields an equivalent (congruent) structure. The process of carrying out the reflection is referred to as the symmetry operation a. However, as the atoms of these molecules are essentially point masses, the reflection operations are in each case simply the inversion of the coordinate perpendicular to the plane of symmetry. Following certain conventions, the reflection operation corresponds to z + z for BF3 and benzene, as it is the z axis that is chq ep perpendicular to die plane, while it is jc —> —x for water. It should be evident that the symmetry operation has an effect on the chosen coordinate systems, but not on the molecule itself. 

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