Rotatory-inversion axis

A rotatory-inversion axis combines a rotation around a line through 3 60°/n with inversion through a specific point on this line. Axes with n = 3, 4, or 6 (designated 3, 4, and 6) are possible in crystals, that with n = 2 is equivalent to a mirror plane, which is perpendicular to the twofold axis and passes through the inversion center. 

FIGURE 4.6. A rotatory-inversion axis involves a rotation and then an inversion across a center of symmetry. Since, by the definition of a point group, one point remains unmoved, this must be the point through which the rotatory-inversion axis passes and it must lie on the inversion center (center of symmetry). The effect of a fourfold rotation-inversion axis is shown in two steps. By this symmetry operation a right hand is converted to a left hand, and an atom at x,y,z is moved to y,—x,—z. (a) The fourfold rotation, and (b) the inversion through a center of symmetry. 

FIGURE 4.6. (c) A rotatory-inversion axis. The view from above, where filled circles lie at +z and open circles at —z. Two steps are involved (1) a fourfold rotation, and (2) inversion about the origin. 

The fourth type of symmetry operation combines rotational symmetry with inversion symmetry to produce what is called a rotatory-inversion axis, designated n (Figure 4.6). It consists of rotation about a line combined with inversion about a specific point on that line. For example, the operation of fourfold rotation-inversion is done by rotating an object at x,y,z through an angle of 90° about the z axis to produce an... 

Triclinic Identity or inversion [onefold rotation or rotatory-inversion axis) in any direction T a h c + + i... 

Rotatory—inversion axis Rotation of an object by 360°/n, about this axis and then inversion through a center of symmetry to give a mirror-image form of the original object. 

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). 

For a finite group of atoms the criterion for enantiomorphism is the absence of an axis of rotatory inversion. An axis n implies a centre of symmetry if n is odd, it introduces planes of symmetry if k is a multiple of 2 but not of 4, and if k is a multiple of 4 the system can be brought into coincidence with its mirror image. Of the simplest axes of these three types, T is synonymous with a centreof symmetry, and 2 with a plane of symmetry. Since axes of rotatory inversion 4 are likely to occur very rarely in molecules, we may for practical purposes take as the criterion for enantiomorphism and for optical activity in a finite molecule or complex ion the absence of a centre or plane of symmetry. 

Not all rotatory-reflections are unfamihar operations. An S is just a C followed by a (T/i - this is equivalent to the mirror reflection alone. S2 is equal to the inversion, because the rotation reverses the signs of the coordinates measured along axes (of the coordinate system) perpendicular to the axis (of rotation), and the reflection reverses the sign of the third coordinate. 

The question of whether the optical inversion is a true chemical inversion has been of concern to us (1,20,21) for sometime, too. The rationale of a chemical inversion being associated with the optical inversion is based on the "nonempirical" analysis of the tt-tt electronic transitions of the tris and bis chelates (22-27). The long-axis tt-it head-to-head A2 and B levels of the tris and bis chelates, respectively, are expected to be at higher energies than the long-axis head-to-tail E and A levels of the tris and bis chelates, respectively. For A isomers, the head-to-head transitions should have negative rotatory strengths and the head-to-tail ones should be positive. Therefore, retention... 

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