Identity element 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. 

Figure 15b represents a dinuclear complex formed by a pair of tetrahedral complexes Af bc and a metallic bond. Such a molecule has only an identity element of symmetry. However, as all the identical ligands are placed exactly opposite to each other, an indistinguishable appearance can be obtained by interchanging the opposite positions, which means that it possesses the centre of inversion i. Hence, this complex is an example of the point group Q. The meso form of the organic compound, CHCIBr—CHCIBr, is another example of the point group Q. 

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. 

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. 

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 ... 

An element of symmetry is possessed by a molecule if, after the associated symmetry operation is carried out, the atoms of that molecule are not perceived to have moved. The molecule is then in an equivalent configuration. The individual atoms may have moved but only to positions previously occupied by identical atoms. 

Prior to interpreting the character table, it is necessary to explain the terms reducible and irreducible representations. We can illustrate these concepts using the NH3 molecule as an example. Ammonia belongs to the point group C3V and has six elements of symmetry. These are E (identity), two C3 axes (threefold axes of rotation) and three crv planes (vertical planes of symmetry) as shown in Fig. 1-22. If one performs operations corresponding to these symmetry elements on the three equivalent NH bonds, the results can be expressed mathematically by using 3x3 matrices. ... 

Symmetry elements provide the basis of symmetry operations. Thus, a molecule A is said to contain a given element of symmetry when the derived symmetry operation transforms A into a molecule to which it is superimposable. Elements and operations of symmetry are presented in Table 2, with the exception of the pseudooperation of identity which will not be considered. Table 2 shows that corresponding elements and operations of symmetry share the same symbol, and indeed these two terms lack independent meaning. 

A symmetry element is defined as an operation that when performed on an object, results in a new orientation of that object which is indistinguishable from and superimposable on the original. There are five main classes of symmetry operations (a) the identity operation (an operation that places the object back into its original orientation), (b) proper rotation (rotation of an object about an axis by some angle), (c) reflection plane (reflection of each part of an object through a plane bisecting the object), (d) center of inversion (reflection of every part of an object through a point at the center of the object), and (e) improper rotation (a proper rotation combined with either an inversion center or a reflection plane) [18]. Every object possesses some element or elements of symmetry, even if this is only the identity operation. 

We consider first groups lacking a C axis. A molecule with no elements of symmetry (other than the identity E) belongs to the point group C. The identity element is equivalent to Ci, a rotation by In/l radians or 360°. Very few small molecules are this unsymmetrical—one example is the trisubstituted methane ... 

Group (mathematical) A collection or a set of symmetry elements that obey certain mathematical conditions that interrelate these elements. The conditions are that one element is the identity element, that the product of any two elements is also an element, and that the order in which symmetry elements are combined does not affect the result. For every element there exists another in the group that is called the inverse of the first when these two are multiplied together, the product is the identity element. Some symmetry elements are their own inverses for example, a twofold axis applied twice gives the same result as the identity operation. 

Molecules that possess certain elements of symmetry are not chiral, because the element of symmetry ensures that the mirror image forms are superimposable. The most common example is a plane of symmetry, which divides a molecule into two halves that have identical placement of substituents on both sides of the plane. A trivial example can be found at any tetrahedral atom with two identical substituents, as, for example, in 2-propanol. The plane subdivides the 2-H and 2-OH groups and the two methyl groups are identical. 

We say that a figure or a molecule has certain elements of symmetry. One or more symmetry operations are associated with each element of symmetry. A symmetry operation may leave some or all parts of the figure in the same position or it may interchange some or all identical parts of the figure. 

Ik) and (I k ) both lie on an element of symmetry. Then application of this symmetry operation (rotation axis or reflection plane) brings the pair of molecules into self-coincidence. This requires the ij(lk, I k ) to be identical before and after transformation, However, a change of sign may be required, and it can then be concluded that the particular force constant must vanish. For example, assume that the symmetry element is a twofold axis parallel to x. Then I k ) will not change sign hut... 

---