Improper axis of symmetry

FIGURE 1.14 The 4(84) inversion (improper) axis of symmetry in the tetrahedral CF4 molecule. 

The equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

If rotation about an axis by 360°ln followed by reflexion through a plane perpendicular to the axis produces an equivalent configuration of a molecule, then the molecule contains an improper axis of symmetry. Such an axis is denoted by Sn, the associated symmetry operation having been described in the previous sentence. The C3 axis of the PC15 molecule is also an S3 axis. The operation of S3 on PC15 causes the apical (i.e. out-of-plane) chlorine atoms to exchange places. 

The operation of reflexion through a horizontal plane may be regarded as a special case of an improper axis of symmetry of order one Sv The rotation of a molecule around an axis by 360° produces an identical configuration (C, = E), and the reflexion in the horizontal plane is the only non-trivial part of the operations associated with the Sx improper axis. This may be symbolized as ... 

If a molecule does not possess an improper axis of symmetry it is termed dissymmetric and cannot have a mirror image that is superposable on... 

Each 5ec-butyl group has a chiral C that can be R or S. Since all four groups are equivalent, the order of writing the designation is immaterial, RRRS is the same as RRSR. The possibilities are RRRR, RRRS, RRSS, SSRR, SSSR, and SSSS. RRRR and SSSS, and RRRS and SSSR, are enantiomeric pairs. These are the four optically active isomers. The mirror image of RRSS is SSRR. These are identical and therefore the isomer is meso. This isomer is a rare example of a compound which is achiral because it has only an improper axis of symmetry—it has no plane or center of symmetry. 

Identity element, 387-388 Identity operation, 54, 395 Improper axis of symmetry, 53 Improper rotation, 396 Index of refraction, 132 INDO method, 71, 75-76 and ESR coupling constants, 380 and force constants, 245 and ionization potentials, 318 and NMR coupling constants, 360 Induced dipole moment, 187 Inertial defect, 224-225 Inertia tensor, 201... 

Chirality is a concept well known to organic chemists and to all chemists concerned in any way with structure. The geometric property that is responsible for the nonidentity of an object with its mirror image is called chirality. A chiral object may exist in two enantiomorphic forms that are mirror images of one another. Such forms lack inverse symmetry elements, that is, a center, a plane, and an improper axis of symmetry. Objects that possess one or more of these inverse symmetry elements are superimposable on their mirror images they are achiral. All objects belong to one of these categories. 

Improper axis of symmetry Rotation through the angle 2%ln followed by reflection in the plane perpendicular to the rotation axis S ... 

In the application of this general definition, it is possible to include an examination of the symmetry group to which a molecule belongs. Knowledge of its symmetry group allows us to say whether a molecule is chiral or not. The condition for a molecule to be chiral is that it has no element of inverse symmetry, that is it does not have a centre, a plane or an improper axis of symmetry (Figure 2.9). 

There are two other types of symmetry elements. The first is called the identity element, represented by E. Everything has as a symmetry operation it is the symmetry operation due to the object s very existence. The last symmetry element is an improper axis of symmetry, indicated by S . (C is more specifically called a proper rotation.) It is a combination of a C rotation (that is, turning on an axis by 360°In) followed by reflection through a plane that is perpendicular to the axis. Figure 13.4 illustrates the S symmetry operation. Sj is equivalent to a o symmetry operation, and 2 is equivalent to a center of inversion, i. The rotational part of the S symmetry element may or may not correspond with an existing axis of symmetry. 

For classes with fewer than four sites, the assertion is trivial. For chiral classes with four or more sites, there is at least one triple of sites which does not lie in a symmetry plane of the skeleton. For, if all sites lie in a common symmetry plane, molecules of the class with the ligands all different would possess planes of symmetry, i.e., the class would not be chiral. On the other hand, suppose that the sites do not lie all in a common mirror plane, but that nevertheless every triple of sites lies in a symmetry plane. It follows that every pair of sites lies on the intersection of two different symmetry planes, therefore on an axis of symmetry of the skeleton. But if more than four sites all lie pairwise on an axis of symmetry of a finite figure, they must all lie on a common axis, and the class is again achiral. For chiral classes, then, there is at least one triple of sites which does not lie on a plane of symmetry of the skeleton. Now consider a molecule in which the sites of this triple are occupied by ligands of three different kinds, the other sites by ligands different from these three, but identical with each other. Such a molecule is chiral, since the only improper operation which leaves the three different ligands invariant is a reflection in the plane of the triple, and this changes the rest of the molecule. The assertion follows immediately. 

This is the operation of clockwise rotation by 2w/ about an axis followed by reflection in a plane perpendicular to that axis (or vice versa, the order is not important). If this brings the molecule into coincidence with itself, the molecule is said to have a n-fold alternating axis of symmetry (or improper axis, or rotation-reflection axis) as a symmetry element. It is the knight s move of symmetry. It is symbolized by Sn and illustrated for a tetrahedral molecule in Fig. 2-3.3.f... 

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. 

Note that the cis isomer lacks an improper axis of rotation and is therefore chiral, but that the trans isomer has a plane of symmetry and will be achiral in the absence of an asymmetric carbon in the phosphine ligand-28 As in the case of the previously encountered cyclopentadienyl complex (page 476), it can be argued whether the coordination number is 5 or 9. In either semantic interpretation these compounds are of considerable interest since isomerism in nine-coordinate complexes Is even less well documented than in those with coordination number 5. 

It was mentioned above that tris(chelate) complexes of the type (Co(en) ],+ lack an improper axis of rotation. As a result, such complexes can exist in either of two enantiomeric forms (or a racemic mixtire of the two). Figure 12.20 illustrates the complex ions (Co(en)j]3+ and (Crfoxy3-. each of which ts chiral with Di symmetry. 

Thus Sj is equivalent to L Confirm this to your satisfaction with tnuLs-N2F2. which contains a center of symmetry and thus must have a two-fold improper axis of rotation. Note that the SiF4 molecule, although it possesses true C2 axes, does not have a center of symmetry, and thus cannot have an S2 axis. Furthermore S, is equivalent to c because, as we have seen, C, = E and therefore the second step, reflection, yields... 

If a molecule possesses one main n-fold axis of symmetry all its symmetry operations must leave the main symmetry axis unaltered or at most, reverse its direction. Apart form rotations or improper rotations about the main axis the only other symmetry operations which satisfy this condition are a reflection in a plane perpendicular to the main axis (such a plane is called a horizontal plane and the reflection operation is denoted by ch a reflection in a plane containing the main axis (such a plane is called a... 

It was mentioned above that tris(chelate) complexes of the type (Co(en) ] lack an improper axis of rotation. As a result, such complexes can exist in either of two enantiomeric forms (or a racemic mixtire the two). Figure 12.20 illustrates the complex iems (Colen),] and (Cr(ox)3p". each of which is chiral irifh >, symmetry. It is not necessary to have three chelate rings present. The cation dichloro-bis(ethylenediamine)coball(I () exists as two geometric isomers, cis and trans The trans isomer has approximate D f, symmetry (Hg. 12.21b). Because it has three internal miiror planes, it is achiral. The cis isomer has symmetry and is chiral (Fig. I2.2la>. Since the two chlmde ions rq)Jace two nitrogen atoms from an eth-... 

S An improper rotation or rotation-reflection axis. Clockwise rotation through an angle of Injn radians followed by a reflection in the plane perpendicular to the axis of rotation. Also known as an alternating axis of symmetry. Note that S is equivalent to a/, and S2 is equivalent to i. 

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