Symmetry elements kinds

Chiral molecules are characterised by symmetry elements of the first kind, for example, axes of rotation. 

Lord Kelvin lla> recognized that the term asymmetry does not reflect the essential features, and he introduced the concept of chiralty. He defined a geometrical object as chiral, if it is not superimposable onto its mirror image by rigid motions (rotation and translation). Chirality requires the absence of symmetry elements of the second kind (a- and Sn-operations) lu>>. In the gaseous or liquid state an optically active compound has always chiral molecules, but the reverse is not necessarily true. 

For each symmetry element of the second kind (planes of reflection and improper axes of rotation) one counts according to Eq. (1) the pairs of distinguishable ligands at ligand sites which are superimposable by symmetry operations of the second kind. 

For the individual symmetry elements of the second kind different values for u may result. The lowest value of u, called wmin, is the number... 

A mathematical group consists of a set of elements which are related to each other according to certain rules, outlined later in the chapter. The particular kind of elements which are relevant to the symmetries of molecules are symmetry elements. With each symmetry element there is an associated symmetry operation. The necessary rules are referred to where appropriate. 

In some circumstances the magnitudes of the translation vectors must be taken into account. Let us demonstrate this with the example of the trirutile structure. If we triplicate the unit cell of rutile in the c direction, we can occupy the metal atom positions with two kinds of metals in a ratio of 1 2, such as is shown in Fig. 3.10. This structure type is known for several oxides and fluorides, e.g. ZnSb20g. Both the rutile and tlie trirutile structure belong to the same space-group type PAjmnm. Due to the triplicated translation vector in the c direction, the density of the symmetry elements in trirutile is less than in rutile. The total number of symmetry operations (including the translations) is reduced to... 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

The two things, symmetry elements and symmetry operations, are inextricably related and therefore are easily confused by the beginner. They are, however, different kinds of things, and it is important to grasp and retain, from the outset, a clear understanding of the difference between them. 

TABLE 3.1 The Four Kinds of Symmetry Elements and Operations Required in Specifying Molecular Symmetry... 

We present here some very general and useful rules about how different kinds of symmetry elements and operations are related. These deal with the way in which the existence of some two symmetry elements necessitates the existence of others, and with commutation relationships. Some of the statements are presented without proof the reader should profit by making the effort to verify them. 

Although we have followed conventional practice—and for general purposes will continue to do so—in setting out four kinds of symmetry elements and operations, a, /, C , and Sny we should note that the list can in principle be reduced to only two C and S . A reflection operation can be regarded as an... 

We have now reached a point of departure in the process of adding further symmetry elements to a C axis. We shall consider (1) the addition of different kinds of symmetry planes to the C axis only, and (2) the addition of symmetry planes to a set of elements consisting of the C axis and the n C2 axes perpendicular to it. In the course of this development it will be useful to have some symbols for several kinds of symmetry planes. In defining such symbols we shall consider the direction of the C axis, which we call the principal axis or reference axis, to be vertical. Hence, a symmetry plane perpendicular to this axis will be called,a horizontal plane and denoted ah. Planes that include the C axis are generally called vertical planes, but there are actually two different types. In some molecules all vertical planes are equivalent and are symbolized av. In others there may be two different sets of vertical planes (as in PtClJ" cf. page 32), in which case those of one set will be called ov and those of the other set crrf, the d standing for dihedral. It will be best to discuss these differences more fully as we meet them. 

This property of the diagrams in Figure 1.6 is called a symmetry property. The axis of rotation is called a symmetry element. There are various kinds of symmetry elements an axis is designated by the letter C. Since this particular axis is an infinite-fold rotation axis, in the sense specified above, it is called a axis. The... 

Figure 1.9 shows the three-dimensional shape of the electron distributions Pmo2px and symmetry element is called a C axis. An orbital with this kind of symmetry is called a 77 orbital. Atomic orbitals of the s type can form only a molecular orbitals atomic... 

These procedures will be successful as long as there is no rotational symmetry axis of order higher than two. If there is a threefold or higher axis, some further complications arise. Because of Rule 6, it will often be possible to make symmetry arguments on the basis of other symmetry elements in cases of this kind, and we shall not need to consider this problem further. 

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. 

Enantiotopic ligands and faces are not interchangeable by operation of a symmetry element of the first kind (Cn, simple axis of symmetry) but must be interchangeable by operation of a symmetry element of the second kind (cr, plane of symmetry i, center of symmetry or S , alternating axis of symmetry). (It follows that, since chiral molecules cannot contain a symmetry element of the second kind, there can be no enantiotopic ligands or faces in chiral molecules. Nor, for different reasons, can such ligands or faces occur in linear molecules, QJV or Aoh )... 

Symmetry elements of the second kind other than ct may generate enantiotopic ligands. Thus compound 42 in Fig. 16 (F and T are enantiomorphic, i.e. mirror-... 

Chirality is the geometric property of a rigid object (or spatial arrangement of points or atoms), which is nonsuperposable on its mirror image such an object has no symmetry elements of the second kind (a mirror plane, a center of inversion, a rotation-reflection axis,. ..). If the object is superposable on its mirror image, the object is described as being achiral. 

The concept of symmetry and chirality in chemistry has a well-defined meaning only in relation to experiment.18 Consider a system of one or more molecules subject to experimental observation. The properties of any such system are invariant with respect to its symmetry operations.42 In Pierre Curie s famous dictum, c est la dissymetrie qui cree le phenomfcne. 43 That is, a phenomenon is expected to exist—and can in principle be observed—only because certain elements of symmetry are absent from the system. It follows that all manifestations of chirality flow from a single source the absence of symmetry elements of the second kind in the group describing the system under observation. Accordingly, if... 

This group has only two symmetry elements E and i. There are not many molecules with this kind of symmetry. Two examples are given in Fig. 6.2.2. 

---