Improper axis

Is there, then, an improper axis S(Note that if n > 2, the n-fold rotation axis C is by convention taken to be the vertical (z) axis). You have replied that there is indeed an axis Sjn. However, are there other binary axes perpendicular to the If not, the symmetry of your molecule is described by one of the groups Ja, (Note that if n is odd, there is a center of inversion). However, this result is subject to doubt, as there are very few molecules of symmetry J ... 

If you have arrived at this point, you have found no improper axis but more than one proper axis of order n Or, you have identified at least one other axis of order n in addition to that collinear with Sin- Now, look for n binary axes perpendicular to C . If you do not find any, the point group is one of the t -type. Otherwise, there are n binary axes and the group is one of type 0. 

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. 

Hexagonal a=yg=90° 7=120° a=b c One sixfold axis or one sixfold improper axis... 

Tetragonal a=yS=7=90° a=b c One fonrfold axis or one fourfold improper axis... 

Improper axis sn Rotate the molecule by 360/n degrees around the improper axis and then reflect the molecule through the plane perpendicular to the improper axis... 

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. 

Improper rotation axis. Rotation about an improper axis is analogous to rotation about a proper synunetry axis, except that upon completion of the rotation operation, the molecule is mirror reflected through a symmetry plane perpendicular to the improper rotation axis. These axes and their associated rotation/reflection operations are usually abbreviated X , where n is the order of the axis as defined above for proper rotational axes. Note that an axis is equivalent to a a plane of symmetry, since the initial rotation operation simply returns every atom to its original location. Note also that the presence of an X2 axis (or indeed any X axis of even order n) implies that for every atom at a position (x,y,z) that is not the origin, there will be an identical atom at position (—x,—y,—z) the origin in such a system is called a point of inversion , since one may regard every atom as having an identical... 

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... 

As noted earlier, point groups with no threefold or higher proper or improper axis have only one-dimensional representations hence a necessary condition for a molecule to have degenerate vibrational modes is that it possess a Cn or an S axis with n> 3. Asymmetric tops have no degenerate vibrational modes. 

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... 

Improper axis One or more repetitions of the sequence rotation followed by reflection in a plane L to the rotation axis... 

An improper rotation may be thought of as taking place in two steps first a proper rotation and then a reflection through a plane perpendicular to the rotation axis. The axis about which this occurs is called an axis of improper rotation or, more briefly, an improper axis, and is denoted by the symbol where again n indicates the order. The operation of improper rotation by 2nln is also denoted by the symbol Sn. Obviously, if an axis C and a perpendicular plane exist independently, then S exists. More important, however, is that an S may exist when neither the Cn nor the perpendicular a exist separately. 

If the improper axis is of even order and [Pg.35]

There is an even-order improper axis, 5I0, as indicated in Figure 3.15, but there are also other unrelated symmetry elements, so the group is not St0. 

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... 

B As we have. seen. Hie formal definition of optical activity is bused upon the absence of an improper axis of rotation. The Iwo definitions arc equivalent. 

Fig. 3 9 A four-fold improper axis (4) operation. The 3 axis is perpendicular to the plane of the paper, A, and A3 (+) are above the plane, A2 and A (—) are below. When a point Aj (jt, y, z coordinates ...

In the Schoenfiies system the improper axis is an axis of rotation-reflection (see page 52). In the Internationa) system the axis of rotatory inversion ( ) is ore of n-fold rotation followed by inversion (see Fig. 3.29). 

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