Equivalent positions

As briefly mentioned in the previous section, equivalent positions (or sites) that are listed in the field No. 8 in Table 1.18 for each crystallographic space group, represent sets of symmetrically equivalent points found in one unit cell. All equivalent points in one site are obtained from an initial point by applying all symmetry operations that are present in the unit cell. The fractional coordinates (coordinate triplet) of the initial (or independent) point are usually marked as x, y, z. 

Any negative coordinate (i.e. y = -0.15) may be converted into a positive coordinate by adding a full translation along the same axis, i.e. y = -0.15 + 1 =0.85. 

When a point (or an atom) is placed on a finite symmetry element that converts the point into itself, the multiplicity of the site is reduced by an integer factor when compared to the multiplicity of the general site. Since different finite S5mimetry elements may be present in the same space group symmetry, the total number of different non-general sites (they are called special sites or special equivalent positions) may exceed one. Contrary to a general equivalent position, one, two or all three coordinates will be constrained in every atom occupying a special equivalent position. 

Both the multiplicity and Wyckoff letter combined together, are often used as the name of the equivalent position. Sometimes when 

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Although all of the nuclear coordinates participate in this kinetic energy operator, and in our previous discussions, all of the nuclear coordinates are expanded, with respect to an equivalent position, in power series of the parameter K, here in the specific case of a diatomic molecule, we found that only the R coordinate seems to have an equilibrium position in the molecular fixed coordinates. This means that actually we only have to, or we can only, expand the R coordinate, but not the other coordinates, in the way that... 

If the same alkyl group occurs more than once as a side chain, this is indicated by the prefixes di-, tri-, tetra-, etc. Side chains are cited in alphabetical order (before insertion of any multiplying prefix). The name of a complex radical (side chain) is considered to begin with the first letter of its complete name. Where names of complex radicals are composed of identical words, priority for citation is given to that radical which contains the lowest-numbered locant at the first cited point of difference in the radical. If two or more side chains are in equivalent positions, the one to be assigned the lowest-numbered locant is that cited first in the name. The complete expression for the side chain may be enclosed in parentheses for clarity or the carbon atoms in side chains may be indicated by primed locants. 

The 12 hydrogen atoms of cyclohexane do not occupy equivalent positions. In the chair conformation six hydrogen atoms are perpendicular to the average plane of the molecule and six are directed outward from the ring, slightly above or below the molecular plane (see Fig. 1.6). Bonds which are perpendicular to the molecular plane are known as axial bonds, and those which extend outward... 

Alexandrite, like ruby, contains Cr ions but they are substituted in the lattice of chrysoberyl, BeAl204. The chromium ions occupy two symmetrically non-equivalent positions which would otherwise be occupied by aluminium ions. In this environment the 2 ground state of Cr is broadened, compared with that in ruby, by coupling to vibrations of the crystal lattice. 

In the genuine low-temperature chemical conversion, which implies the incoherent tunneling regime, the time dependence of the reactant and product concentrations is detected in one way or another. From these kinetic data the rate constant is inferred. An example of such a case is the important in biology tautomerization of free-base porphyrines (H2P) and phtalocyanins (H2PC), involving transfer of two hydrogen atoms between equivalent positions in the square formed by four N atoms inside a planar 16-member heterocycle (fig. 42). 

Interference equivalent Positive or negative response caused by a substance other than the one being measured. 

To answer the first question, Lesk and Chothia examined in detail residues at structurally equivalent positions that are involved in helix-heme contacts and in packing the a helices against each other. After comparing the nine globin structures then known, the 59 positions they found that fulfilled these criteria were divided into 31 positions buried in the interior of the protein and 28 in contact with the heme group. These positions are the principal determinants of both the function and the three-dimensional structure of the globin family. 

Like Thr 124 and Thr 215, the Asn 69 and Asn 159 residues occupy equivalent positions in the two homologous motifs of TBP. By analogy with the symmetric binding of a dimeric repressor molecule to a palindromic sequence described in Chapter 8, the two motifs of TBP form symmetric sequence-specific hydrogen bonds to the quasi-palindromic DNA sequence at the center of the TATA box. The consensus TATA-box sequence has an A-T base pair at position 4, but either a T-A or an A-T base pair at the symmetry-related position 5, and the sequence is, therefore, not strictly palindromic. However, the hydrogen bonds in the minor groove can be formed equally well to an A-T base pair or to a T-A base pair, because 02 of thymine and N3 of adenine occupy nearly stereochemically equivalent positions, and it is sufficient, therefore, for the consensus sequence of the TATA box to be quasi-palindromic. 

A here y is the number of equivalent positions. A partial rate factor calculation for nitration of toluene is given in Example 10.1. 

In principle, this aspect of the reactivity of aza-activated compounds might be deduced from data of the kind shown in Table VII, but experimentally it is easier to obtain by direct comparison of the mobilities of the leaving group from the non-equivalent positions of a given... 

Relative reactivities at non-equivalent positions are a function of the structure of the substrate as well as of the reagent type, which should be allowed for in the theoretical treatment. 

Benzimidazole Equivalent position /fj- = 1 is little affected by substitution in... 

Surprinslngly, we observe an drastic effect of the concentration on the SRO contribution (figure 2) indeed, in PtaV, the maxima are no longer located at a special point of the fee lattice but the (100) intensity is splltted perpendicularly in the (010) direction and presents a saddle point at (100) position. Notice that these two maxima are not located just above Bragg peaks of the ordered state the A B ground state presents Bragg peaks at ( 00) and equivalent positions whereas the SRO maxima peak between ( 00) and (100). 

According to crystal analysis performed by Stomberg [173], Na2NbOF5 is made up of sodium ions and isolated NbOF52 complex ions and is similar in structure to FeWC>6. NbOFs2" polyhedrons comprise slightly distorted octahedrons that are located in one of two equivalent positions. The niobium atom is shifted 0.234 A from the equatorial plane towards the oxygen atom. 

Modification of filler s surface by active media leads to the same strong variation in viscosity. We can point out as an example the results of work [8], in which the values of the viscosity of dispersions of CaC03 in polystyrene melt were compared. For q> = 0.3 and the diameter of particles equal to 0.07 nm a treatment of the filler s surface by stearic acid caused a decrease in viscosity in the region of low shear rates as compared to the viscosity of nontreated particles more than by ten times. This very strong result, however, should not possibly be understood only from the point of view of viscometric measurements. The point is that, as stated above, a treatment of the filler particles affects its ability to netformation. Therefore for one and the same conditions of measuring viscosity, the dispersions being compared are not in equivalent positions with respect to yield stress. Thus, their viscosities become different. 

Two compounds can be produced. Resonance makes positions I, 4, 6, and 9 equivalent. It also makes positions 2, 3, and 8 equivalent. Positions 5 and 10 are equivalent but have no H atom. 

Atomic site Number of equivalent positions Occupation probabilities ... 

There are no known examples of supported clusters dispersed in crystallo-graphically equivalent positions on a crystalline support. Thus, no structures have been determined by X-ray diffraction crystallography, and the best available methods for structure determination are various spectroscopies (with interpretations based on comparisons with spectra of known compoimds) and microscopy. The more nearly uniform the clusters and their bonding to a support, the more nearly definitive are the spectroscopic methods however, the uniformities of these samples are not easy to assess, and the best microscopic methods are limited by the smallness of the clusters and their tendency to be affected by the electron beam in a transmission electron microscope furthermore, most supported metal clusters are highly reactive and... 

In systems such as [A... A ]+ where an electron (or a hole) hesitates or oscillates between two equivalent positions on subsystems A or A, symme breakings may occur when the effective transfer integral between the two sites is weak. Hiis will be the case when A and A are far apart, when they are bridged by an "insulating" ligand, or when the two localized MOs concerned by the electron transfer have a very we spatial overlap. 

Symmetrically equivalent positions -x,-y,z -y,x,z y,-x,z 2.9 Calculate the Zr-O bond lengths in baddeleyite (Zr02), considering only interatomic distances shorter than 300 pm. What is the coordination number of Zr ... 

When not all points are equivalent, the numbers of equivalent positions are given as sums... 

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 the same kind tend to be in equivalent positions. 

The given conditions do not always allow for equivalent positions for all atoms. Take as an example the following conditions composition MX5, covalent M-X bonds, all X atoms bonded to M atoms. In this case all X atoms can only be equivalent if each set of five of them form a regular pentagon around an M atom (as for example in the XeFj ion). If this is not possible for some reason, then there must be at least two non-equivalent positions for the X atoms. According to the symmetry principle the number of these non-equivalent positions will be as small as possible. 

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. 

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