Rotation axis with intersecting symmetry

A Rotation Axis with Intersecting Symmetry Planes... 

In addition to a rotation axis with intersecting symmetry planes (which is equivalent to having multiple intersecting symmetry planes), snowflakes have a perpendicular symmetry plane. This combination of symmetries is labeled m-n.m and it is characteristic of many other... 

There are several forms of rotational symmetry. The simplest is cyclic symmetry, involving rotation about a single axis (Fig. 4—24a). If subunits can be superimposed by rotation about a single axis, the protein has a symmetry defined by convention as Gn (C for cyclic, n for the number of subunits related by the axis). The axis itself is described as an w-fold rotational axis. The a/3 protomers of hemoglobin (Fig. 4-23) are related by C2 symmetry. A somewhat more complicated rotational symmetry is dihedral symmetry, in which a twofold rotational axis intersects an w-fold axis at right angles. The symmetry is defined as DTO (Fig. 4—24b). A protein with dihedral symmetry has 2n protomers. 

In Section VIII we described a method for finding the most probable rotationally symmetric shape given measurements of point location. The solution for mirror symmetry is similar. In this case, given m measurements (where m - 2q), the unknown parameters are fyjpj, (0 and 0 where 0 is the angle of the reflection axis. However these parameters are redundant and we reduce the dimensionality of the problem by replacing two-dimensional (0 with the one dimensional x0 representing the x-coordinate at which the reflection axis intersects the x-axis. Additionally we replace Rt, the rotation matrix with ... 

It might seem possible to extend the group still further by having both sets of symmetry planes simultaneously present. In this case, however, one would have a set of planes intersecting along the C2 axes at angles of 45. The product of two reflections in adjacent planes would be a rotation C4 about their line of intersection which would, therefore, be a symmetry axis with n = 4 and would belong to the next case to be described. 

Fivefold symmetry appears frequently among primitive organisms. Examples are shown in Figure 2-19. They have fivefold rotation axes and intersecting (vertical) symmetry planes as well. The symmetry class of the starfish is 5 m. This starfish consists of ten congruent parts, with each pair related by a symmetry plane. The whole starfish is unchanged either by 360°/5 = 12° rotation around the rotation axis, or by mirror reflection through the symmetry planes which intersect at... 

The twofold mirror-rotation axis is the simplest among the mirror-rotation axes. There are also axes of fourfold mirror-rotation, sixfold mirror-rotation, and so on. Generally speaking, a 2 -fold mirror-rotation axis consists of the following operations a rotation by (360/2//) and a reflection through the plane perpendicular to the rotation axis. The symmetry of the snowflake involves this type of mirror-rotation axis. The snowflake obviously has a center of symmetry. The symmetry class m-6 m contains a center of symmetry at the intersection of the six-fold rotation axis and the perpendicular symmetry plane. In general, for all m n m symmetry classes with n even, the point of intersection of the //-fold rotation axis and the perpendicular symmetry plane is also a center of symmetry. When n is odd in an m-n m symmetry class, however, there is no center of symmetry present. 

When the quadratic coupling terms in Eq. (3) are included, the rotational symmetry of the potential surfaces in Eq. (4) is lost and replaced by a threefold symmetry inherent to systems with a threefold rotation axis. Consequently, j ceases to be a good quantum number and the spectra of the linear E e JT system for individual j become mixed . This inherently two-dimensional vibronic motion leads to a complicated, erratic line structure (see, for example, Ref. 21) as is typical for other, less symmetric conical intersections discussed throughout this book. The above statements about adiabatic and nonadiabatic behavior for i < 0 and E > 0, and the formation of broad quasi-resonances arising from the upper cone vibrational levels, are not affected by the inclusion of quadratic coupling terms. 

Which plane is which is a somewhat arbitrary choice however, the designation described here is quite widely used and is based on the alignment of the symmetry planes with the Cartesian axis system. Figure 1.11 shows that the two planes intersect in the line of the C2 axis we identified earlier the planes of symmetry are said to contain the rotational axis. The principal axis gives us the Cartesian Z-direction, which, in this case, is in... 

For example, in a plane triangular molecule such as BF3, each of the twofold symmetry axes lying in the plane can be carried into coincidence with each of the others by rotations of 27r/3 or 2 x 2nl3, which are symmetry operations. Thus all three twofold axes are said to be equivalent to one another. In a square planar AB4 molecule, there are four twofold axes in the molecular plane. Two of them, C2 and C2, lie along BAB axes, and the other two, C and Ci, bisect BAB angles. Such a molecule also contains four symmetry planes, each of which is perpendicular to the molecular plane and intersects it along one of the twofold axes. Now it is easy to see that C2 may be carried into C2 and vice versa, and that C2 may be carried into C2 and vice versa, by rotations about the fourfold axis and by reflections in the symmetry planes mentioned, but there is no way to carry C2 or C into either CJ or Cn or vice versa. Thus C2 and C2 form one set of equivalent axes, and and C form another. Similarly, two of the symmetry planes are equivalent to each other, but not to either of the other two, which are, however, equivalent to each other. 

The group Dy, can be generated by only three symmetry elements, namely, the three planes of symmetry. Each pair of reflections in the planes generates a C2 rotation about the axis formed by the intersection of the planes thus the symmetry properties of a wave function with respect to the planes will automatically determine its symmetry with respect to the rotations, and the latter need not be explicitly considered. 

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