Improper Rotation, Sn

The fifth type of point symmetry operation is an improper rotation, given the symbol S . Improper rotations occur around an improper rotational axis by rotating the molecule by 2rr/n radians and then reflecting the molecule through a mirror 

The inversion center in benzene lies in the plane of the molecule directly in the center of the ring. 

Stepwise illustration of an Sg rotation, showing how an improper rotation consists of a rotation around the axis by 2jcI6 radians, followed by reflection through a perpendicuiar plane. 

TABLE 8.1 The different types of point symmetry elements and operations. 

Symmetry Operation Symmetry Element What Happens  

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A molecule can rotate the plane of polarised light only if it cannot be superimposed on its mirror image. The symmetry element to look for is an axis of improper rotation Sn if one is present the object can be superimposed on its mirror image, and therefore cannot be optically active. If Sn is absent, the superposition is impossible and molecule maybe optically active. 

Improper rotation Sn Improper rotation axis (= point of intersection of a proper rotation axis and a perpendicular reflection plane) 0... 

Proper rotations are identified in character tables by the symbols Cn, improper rotations, Sn, are all other operations in the group, the special improper rotations corresponding to inversion and reflection, being identified separately by the symbols i, Oy, Od<... 

Improper rotation (Sn). Improper rotation is a rotation by 2nln followed by reflection through a plane perpendicular to the rotation axis. An axis of improper rotation Sf) is an axis about which this operation leaves the structure of a molecule unchanged. 

An electric dipole operator, of importance in electronic (visible and uv) and in vibrational spectroscopy (infrared) has the same symmetry properties as Ta. Magnetic dipoles, of importance in rotational (microwave), nmr (radio frequency) and epr (microwave) spectroscopies, have an operator with symmetry properties of Ra. Raman (visible) spectra relate to polarizability and the operator has the same symmetry properties as terms such as x2, xy, etc. In the study of optically active species, that cause helical movement of charge density, the important symmetry property of a helix to note, is that it corresponds to simultaneous translation and rotation. Optically active molecules must therefore have a symmetry such that Ta and Ra (a = x, y, z) transform as the same i.r. It only occurs for molecules with an alternating or improper rotation axis, Sn. 

A twofold rotation around the molecular axis of cA-7,2-dichloroethylene is not a covering operation, because the rotation exchanges hydrogens and chlorines. However, the compound operation of the twofold rotation followed by a reflection in a plane perpendicular to the molecular axis - a C2 followed by a ah — is a covering operation. The combination of an n—fold rotation and a reflection in a perpendicular plane is called a rotatory-reflection or improper rotation, symboUzed Sn-... 

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... 

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. 

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. 

This denotes an axis about which a rotation-reflection (or improper rotation) operation may be carried out. The rotation-reflection operation Sn involves a... 

PROBLEM 7.3.2. Prove that molecules can be chiral (= optically active) if and only if they do not have an Sn improper rotation axis (n > 1) [1, p. 435]. 

Improper Rotations A rotation by 360/n about an axis followed by a reflection in a plane perpendicular to the axis is called rotation-reflection symmetry operation. A combined operation of this kind is called a rotation-reflection or an improper rotation and is denoted by the symbol Sn standing for the combination of a rotation through an angle 2%/n about some axis and reflection in a plane perpendicular to the axis. C4 operation followed by reflection through the plane of molecule gives S4 axis. If we use the symbol oh to denote the reflection in the plane perpendicular to rotatory-reflection axis we can write... 

One should be careful of any improper rotation axes that may be implied by the group rather than just written explicitly. For example, the group Cnh include sn i n a concealed form because they include Cn and on. Any group containing the inversion as an element also possess at least the element S,y because an inversion can be envisaged as a 180 rotation followed by a ah reflection. It follows that molecules with centres of inversion cannot be optically active. However, not all molecules without a centre of inversion are optically active. For instance, if their symmetry is S4 they lack an i element but possess S4 which implies inactivity. 

Sn improper rotation through liijn, that is, C followed by Oh... 

The S 2n groups (n = 1, 2, 3), with additional rotations jt/n about the main axis, followed by a reflection through a plane perpendicular to the main axis ( n or S n-1 rotation-reflections). For n = 1, this corresponds to inversion /. The Sn operations are called improper rotations, by comparison with the proper rotations Gn. The only element of group S2 (besides E) is / so that this group is also noted G . 

Sn Improper rotation axis. This involves sequential steps of rotation by 360°/n,... 

Sn = C/, where m runs from 1 to (n - 1) and p = 2mfor 2m < n or p = 2m - n for 2m > n that is, an even number of applications of an odd-order improper rotation is the same as a related simple rotation. The operations with rotations less than 360° can also be described as the same number of simple rotations, and improper rotations involving angles greater than 360° are related to odd numbers of simple rotations. 

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. 

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... 

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