Axis of symmetry, rotation about

To treat the reagents and products as indistinguishable, one must make the total (electronic + nuclear) wave function symmetric under a cyclic exchange of nuclei, which is equivalent to making it symmetric under rotations 2n/3, 4n/3. about the threefold axis of symmetry. Mead showed that, because the electronic wave function >]> is antisymmetric under 2ti/3> must be symmetrized... 

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

The type of symmetry present in each type of metallocene initiator (C2v, C2, Cs, Ci) is listed in Table 8-5. The symmetry elements (axis and plane) for each type is indicated. An axis is a C2 axis of symmetry when rotation of 180° about that axis yields a structure indistinguishable from the original structure. The stereoselectivity of each of the two coordination... 

In Figure 1.12(a) the rotational axis is shown as a vertical line through the O atom in OF2 rotation about this line by 180° in the direction of the arrow, produces an identical looking molecule. The line about which the molecule rotates is called an axis of symmetry, and in this case, it is a twofold axis because we have to perform the operation twice to return the molecule to its starting position. 

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 symmetry elements of the water molecule are easily detected. There is only one proper axis of symmetry, which is the one that bisects the bond angle and contains the oxygen atom. It is a C2 axis and the associated operation of rotating the molecule about the axis by 180° results in the hydrogen atoms exchanging places with each other. The demonstration of the effectiveness of the operation is sufficient for the diagnosis of the presence of the element. 

Axes of SYMMETRY. An axis of symmetry is a line such that rotation of the crystal about this line through an angle 360°/ put the crystal into a position which is indistinguishable from its original position. The value of V is the degree of the axis and can only be 1,... 

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. 

A C axis is often called a proper rotational axis and the rotation about it a proper rotation. An improper rotation may be visualized as occurring in two steps rotation by 360E/ followed by reflection across a plane perpendicular to the rotational axis. Neither the axis of rotation nor the mirror plane need be true symmetry elements that can stand alone. For example, we have seen that SiF4 has C3 axes but no C4 axis. Nevertheless, it has three S4 axes, one through each pair of opposite faces of the cube below ... 

Fig. 3.18 Effects of symmetry operations in C2l. symmetry rotation about the z axis (a) identity. E (b) rotation about the C2 axis, (c, d) reflection in p, planes.

Symmetry mill respect lo a line. If during a complete revolution of 360° about a given axis, a geometrical figure repeats itself in appearance two or more limes, it is said to be symmetrical with respect to a line, or to an axis of symmetry. Possible axes of symmetry are twofold, threefold, fourfold, and sixfold. A rotation of onefold or 360° is equivalent to no rotation at all. 

Now the molecular orbitals in equation (3) also have interesting transformation properties. Consider the effect of a rotation through 2w/5 radians about the fivefold axis of symmetry, perpendicular to the plane of the ring. For example, in place of 0+i take that function which is obtained from it by replacing atom 5 (and thus 0b) by atom 4 (and thus 04), 4 by 3, and so on, namely... 

It may be useful to illustrate this idea with one or two examples. The H2 molecule (or any other homonuclear diatomic) has cylindrical symmetry. An electron that finds itself at a particular point off the internuclear axis experiences exactly the same forces as it would at another point obtained from the first by a rotation through any angle about the axis. The internuclear axis is therefore called an axis of symmetry we have seen in Section 1.2 that such an axis is called an infinite-fold rotation axis, CFigure 10.2 illustrates the Cm symmetry and also some of the other symmetries, namely reflection in a mirror plane, abbreviated internuclear axis and equidistant from the nuclei, and rotation of 180° (twofold axis, C2) about any axis lying in that reflection plane and passing through the internuclear axis. (There are infinitely many of these C2 axes only two are shown.) There are, in addition to those elements of symmetry illustrated, others an infinite number of mirror planes perpendicular to the one illustrated and containing the internuclear axis, and a point of inversion (abbreviated i) on the axis midway between the nuclei. 

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