Symmetry operations: of the first kind

Scheme 1. Comparison of whole molecules to determine isomeric relationships. The question marks signify Superposition yes or no . The three tests are Syi —comparison by symmetry operations of the first kind (rotation, torsion) BG—comparison of bonding (connectivity) graphs vertex by vertex Syn—comparison by symmetry operations of the second kind (reflection).

The operators of discrete rotational groups, best described in terms of both proper and improper symmetry axes, have the special property that they leave one point in space unmoved hence the term point group. Proper rotations, like translation, do not affect the internal symmetry of an asymmetric motif on which they operate and are referred to as operators of the first kind. The three-dimensional operators of improper rotation are often subdivided into inversion axes, mirror planes and centres of symmetry. These operators of the second kind have the distinctive property of inverting the handedness of an asymmetric unit. This means that the equivalent units of the resulting composite object, called left and right, cannot be brought into coincidence by symmetry operations of the first kind. This inherent handedness is called chirality. 

Symmetry operations of the first kind and of the second kind are sometimes distinguished in the literature (cf. Ref. (2-40)). Operations of the first kind are sometimes also called even-numbered operations. For example, the identity operation is equivalent to two consecutive reflections from a symmetry plane. It is an even-numbered operation, an operation of the first kind. Simple rotations are also operations of the first kind. Mirror rotation leads to figures consisting of right-handed and left-handed components and therefore is an operation of the second kind. Simple reflection is also an operation of the second kind as it may be considered as a mirror rotation about a onefold axis. A simple reflection is related to the existence of two enan-tiomorphic components in a figure. Figure 2-52 illustrates these distinctions by a series of simple sketches after Shubnikov [2-40]. In accordance with the above description, chirality is sometimes defined as the absence of symmetry elements of the second kind. 

Figure 2-52. Examples of symmetry operations of the first kind (a) and of the second kind (b), after Shubnikov [2-40].

Are they related by a symmetry operation of the first kind ... 

A symmetry operation of the first kind is a proper rotation (C ), as opposed to an improper rotation (S ). 

Enantiotopic ligands and faces are not interchangeable by operation of a symmetry element of the first kind (Cn, simple axis of symmetry) but must be interchangeable by operation of a symmetry element of the second kind (cr, plane of symmetry i, center of symmetry or S , alternating axis of symmetry). (It follows that, since chiral molecules cannot contain a symmetry element of the second kind, there can be no enantiotopic ligands or faces in chiral molecules. Nor, for different reasons, can such ligands or faces occur in linear molecules, QJV or Aoh )... 

Since handedness (left-handed versus right-handed) is important in molecules the eight symmetry operations can be rethought as (C) 4 operations of the first kind (which preserve handedness) translation, identity, rotation, and screw rotation (D) 5 operations of the second kind (which reverse handedness, and produce enantiomorphs) inversion, reflection, rotoinversion, and glide planes. 

Hie Tetrahedron. We consider first a regular tetrahedron. Figure A5-6 shows some of the symmetry elements of the tetrahedron, including at least one of each kind. From this it can be seen that the tetrahedron has altogether 24 symmetry operations, which are as follows ... 

There are five kinds of symmetry operations that one can utilize to move an object through a maximum number of indistinguishable configurations. One is the trivial identity operation E. Each of the other kinds of symmetry operation has an associated symmetry element in the object. For example, our ammonia model has three reflection operations, each of which has an associated reflection plane as its symmetry element. It also has two rotation operations and these are associated with a common rotation axis as symmetry element. The axis is said to be three-fold in this case because the associated rotations are each one-third of a complete cycle. In general, rotation by iTt/n radians is said to occur about an -fold axis. Another kind of operation—one we have encountered before is inversion, and it has a point of inversion as its symmetry element. Finally, there is an operation known as improper rotation. In this operation, we first rotate the object by some fraction of a cycle about an axis, and then reflect it through a plane perpendicular to the rotation axis. The axis is the symmetry element and is called an improper axis. 

Fig. 12 Le/i Effect of the flow configuration and methane conversion fraction (PR) on the stress. Case of an anode-supported cell with LSM-YSZ cathode and compressive gaskets, a Temperature profile and b First principal stress in the anode. The MIC is displayed in transparency, c First principal stress in the cathode (insert alxtve the symmetry line), d Contact pressure on the cathode GDL and compressive gasket and e vertical displacement along the z-axis, with an amplification factor of 2,000. Right column effect of creep in a cell based on a LSCF cathode and a temperature distribution, on b the evolution of the first principal stress in the anode support in operation and c during thermal cycling to RT and d evolution of the first principal stress in the GDC compatibility layer after thermal cycling. The profiles above and below the symmetry axis refer to different operation time [88, 89]. Reproduced here with kind permission from Elsevier 2012...

As a more complex example, in which all four types of symmetry operation and element are represented, let us take the Re2Gi ion, which has the shape of a square parallepiped or right square prism (Fig. A5-4). This ion has altogether six axes of proper rotation, of four different kinds. First, the Rei Re2 line is an axis of fourfold proper rotation, C4, and four operations, C4, C4, C4, C4 = E, may be carried out. This same line is also a C2 axis, generating the operation C2. It will be noted that the Cl operation means rotation by 2 x 2ir/4, which is equivalent to rotation by 2ir/2, that is, to the C2 operation. Thus the C2 axis and the C2 operation are implied by, not independent of, the C4 axis. There are, however, two other types of C2 axis that exist independently. There are two of the type that passes through the centers of opposite vertical edges of the prism, C2 axes, and two more that pass through the centers of opposite vertical faces of the prism, C axes. 

The 5 operation may seem an arbitrary kind of operation, but it must be included as one of the kinds of symmetry operations. For example, the transformation from the first to the third CH4 configuration in Fig. 12.5 certainly meets the definition of a symmetry operation, but it is neither a proper rotation nor a reflection nor an inversion. 

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