Symmetry equivalent positions

Examples for translationengleiche group-subgroup relations left, loss of reflection planes right, reduction of the multiplicity of a rotation axis from 4 to 2. The circles of the same type, O and , designate symmetry-equivalent positions... 

Atoms of an element in symmetry-equivalent positions are substituted by several kinds of atoms. For example CC (diamond) -t ZnS (zinc blende). 

Symmetry-equivalent positions split into several positions that are independent of one another. 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

For other centric space groups, the most convenient way to derive the covariance between p vA) and p(rB) is to assume that the densities are calculated as for Pi, and then averaged over the n symmetry-equivalent positions. This leads, for the averaged density pobs, to... 

The Tyrosine Radicals. - In PS II two tyrosine radicals Y,D and Y z are known. The position of YD and Yz within PS II has been determined by X-ray crystallography 7,19,341 see Fig. 1. Yz is located in between P680 and the tetranuclear Mn-cluster (Mm) in the D1 protein. It is involved in ET between these 2 components and is only observed as a transient radical or in the form of a split signal (see below). YD is found in a symmetry equivalent position to Yz in a hydrophobic pocket of the D2 protein. The Y D is stable in the dark for hours. Y D undergoes slow redox reactions with the lower S states of the OEC (see 4.7) in the dark but is not involved in the main ET pathway that leads to water oxidation. Its detailed function is not fully understood. 

We now turn to some fluxional molecules these feature dynamics in which atoms interchange between symmetry-equivalent positions, without bond breaking necessarily occurring. 

Figure 14 Face-centered cubic (fee) packing of crystaUine Ceo-(a) At room temperature the Cgo molecules rotate essentially isotropically, but below 260 K a phase transition occurs from fee to simple cubic, in which the Cgo molecules jump between symmetry equivalent positions, (b) At 5 K all rotation is frozen and the resulting inter-Cso contact is between a six-six ring fusion and a pentagonal face. (Reprinted with permission of Macmillan from W.I.F. David et al. )...

Figure 3. Graphical presentation of amino acid residue packing in the common part of the interdomain interface (see Fig 1 and 2a). Asterisks identify the symmetry equivalent-positions fitxn the C-temiinal domain. The 2-fold symmetry axis is shown in the center of the interface. The surface residues from the N- and C-terminal domains are related by that symmetry and form two nearly parallel layers of residues. Four central residues in each layer form a figure rectangle with approximately 5x8 A sides joint by angles of about 80° and 110°.

Ultimately we want to know how a crystal diffracts X rays and produces the diffraction pattern that it does, and conversely, how the diffraction pattern can be used to reconstruct the crystal. It will be found useful in this regard to consider the crystal as the combination, or product of two distinct components, or functions. The first of these is the contents of a unit cell, characterized mathematically by the coordinates of the atoms in an asymmetric unit along with their space group symmetry equivalent positions. The second is a point lattice that describes the periodic distribution of the unit cell contents, and is characterized by a, b, and c. A crystal may then be concisely defined as the first component, or function, repeated in identically the same way at every nonzero point of the second. This physical process of repetitive superposition is termed a convolution. It can be formulated mathematically as the product of the two components, or functions as... 

We might note in passing that were the Fourier equation applied to asymmetric units related by space group symmetry in a crystallographic unit cell, the expressions for symmetry equivalent atomic positions assume considerable value. Their application can reduce the number of terms in the summation by the number of symmetry equivalent positions. We need, in practice, to consider only the atoms comprising a single asymmetric unit in the actual calculations. 

Next, the Harker sections of the Patterson map are contoured. An example is shown in Figure 9.11. As for electron density maps, this is done by drawing contours, lines of equal density, at regular intervals around specified values of the Patterson function, as discussed in Chapter 10, to obtain a topographical map of the Patterson density. In this manner the major peaks on the section are defined. Remember, however, that a peak in the Patterson map actually represents the end of a vector from the origin of the Patterson map, and this vector, when it appears on a Harker section, is the vector between some atom at x, y, z and the corresponding atom at a symmetry equivalent position. 

The third part of the spin-Hamiltonian describes the interaction of the electron spin with the surrounding nuclear spins. For the a-protons in pentacene (the protons bound to the carbons in positions a, y, s and symmetry equivalent positions, see Fig. 1) the principal axes of the hyperfine tensor coincide with the zero-field tensor axes and the Hamiltonian describing the hyperfine interaction for such a proton can be written Hhf = lAxlSxIx + diAyySyly + SjA SJ. Here Axx = —91 MHz, Ayy = 29 MHz, and A.z = — 61 MHz are empirically determined constants. We have... 

Figure 2.27 Unit cells containing (A) 16 molecules of Lenk (Lenk-1), (B) eight molecules of Lenk (Lenk-2) and (C) four molecules of Menk. Colours represent symmetry equivalent positions. Reprinted from Ref. [96], Copyright 2014 American Chemical Society.

If an operation moves an atom to a symmetry-equivalent position, then all basis vectors to do with that atom will give rise to only off-diagonal elements in the transformation matrix and so contribute zero to the character for the operation. 

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