Symmetry elements and operations

We begin our discussion with the idea of a symmetry operation, which is a process that generates a configuration indistinguishable from the initial one. In total there are five different types of symmetry operation for a single object such as a molecule, namely rotation, inversion, reflection, rotation-reflection, and identity. While the symmetry operation describes the process, the symmetry element describes the property that the molecule must possess in order for that operation to be performed. For example, the symmetry operation rotation requires that the moleeule possesses the symmetry element an axis of rotation , the operation inversion requires the molecule to possess the element inversion center , and so on. There is a proposed convention that the symmetry operation should be written in an italic font and the element in an upright (Roman) font, so, for example, you can perform a C2 operation around a C2 axis, and so on. 

Inversion centers relating equivalent atoms in (a) (SiBrClH)2, (b) Xep4, and (c) Mo(CO)6- 

Rotation-reflection operations in (a) SiCfi, drawn looking down the S4 axis note that in this example neither the rotation hy 2tt/4 hy itself nor the reflection in a plane perpendicular to this axis generate a configuration equivalent to the initial one, hut the combined operation does (b) Sn(T -C5Ph5)2, showing an almost perfect Sio operation, looking down the axis. Reprinted with permission from [2]. Copyright 1984 American Chemical Society. 

For completeness we must also define the identity operation ( leave alone ) as a symmetry operation, which applies to all molecules, so that strictly speaking we should never say that a molecule has no symmetry. This is given the symbol E (in some texts as /). Note that this set of five symmetry operations contains some hidden duplications a can also be described as 5i, and i as 2, while the identity operation is equivalent to Ci so all five symmetry operations can actually be described by either C or S . The various symmetry operations and elements, along with their conventional symbols (known as the Schoenflies system) are summarized in Table 2.1. 

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The simplest symmetry operations and elements needed to describe unitcell symmetry are translation, rotation (element rotation axis), and reflection (element mirror plane). Combinations of these elements produce more complex symmetry elements, including centers of symmetry, screw axes, and glide planes (discussed later). Because proteins are inherently asymmetric, mirror planes and more complex elements involving them are not found in unit cells of proteins. All symmetry elements in protein crystals are translations, rotations, and screw axes, which are rotations and translations combined. 

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. 

Fig. 1. Symmetry operations and elements illustrated for SIf- The effect of each operation is shown by the numbering of the F atoms.

TABLE 1. Symmetry Operations and Elements Symmetry element ... 

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