Multiplicity, 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... 

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

The Wyckoff notation for a set of equivalent positions consists of two parts (i) the multiplicity M, which is the number of equivalent positions per unit cell, and (ii) an italicized small letter a starting at the bottom of the list and moving upward in alphabetical order. For a primitive unit cell, M is equal to the order of the point group from which the space group is derived for centered cells, M is the product of the order of the point group and the number of lattice points per unit cell. 

Multiplicity, Wyckoff letter, site symmetry, and coordinates given for each set of equivalent positions. 

This table is not complete equivalent positions with multiplicities > 32 are not listed. 

Most of the concepts of group theory can readily be demonstrated in terms of operations on a pointer that moves over the dial of a clock i. e. translations on the unit circle. The one-dimensional symmetry that relates operations on an elementary dial with four allowed equivalent positions E, A, B, C among which the pointer may move by discontinuous clicks, represents a group with four elements. The same symbols that identify the fixed points on the dial may also serve to represent the group operations corresponding to clockwise rotations of n7r/2 for n = 1(A), 2(B), 3(C), 4(E). The composition (product) of any two operations corresponds to successive rotations according to the following multiplication table ... 

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... 

In the previous section, an atom or a motif was allocated to a position, (x, y), which was anywhere in the unit cell. Such a position is called a general position, and the other positions generated by the symmetry operators present are called general equivalent positions. The number of general equivalent positions is given by the multiplicity. Thus, the general position (x, y) in the plane group p2, (number 2) has a multiplicity... 

Multiplicity Wyckoff letter Site symmetry Coordinates of equivalent positions... 

We now calculate the 56 vectors between the eight equivalent positions of Pbca which should correspond to the highest maxima in the Patterson function. Note that the corresponding space group of the Patterson function is Pmmm. The general position of this group also has a multiplicity of 8,... 

Similarly to NEUl, NEU2 contains multiple Asp-box motifs of which one is less conserved (residues 247-254) and two are eanonical Asp-boxes (residues 129-136 and 199-206). Stmcturally, Asp boxes show a (3-hairpin structure stabilized by a water molecule at the center of three eonserved residues of serine, aspartate, and tryptophan, and are found in topologically equivalent positions in all members of the sialidase gene family. Interestingly, Asp-boxes are found in protein families having different sequences and three-dimensional structures, sueh as bacterial ribonucleases, reelin,... 

Z is the number of near neighbors of a vacancy, because a bivacancy has Z equivalent positions, which gives us the multiplicity coefficient Z for the vibrational partition function - e.g. 6 for a cubic centered lattice. 

Therefore, the LFR should arise from the vibration of the entire nanosolid interacting with the host matrix. For a freestanding nanosolid, the LFTi corresponds to intercluster interaction. The optical mode is the relative motion of the individual atoms in a complex unit cell that contains more than one atom. For elemental solids with a simple such as the fee structure of Ag, there presents only acoustic modes. The structure for silicon or diamond is an interlock of two fee structures that contain in each cell two atoms in non-equivalent positions, so there will be three acoustic modes and three optical modes. The complex structure of compound ensures multiple optical modes. 

Of course, moving the Hquid feed and withdrawal positions continuously is impractical. However, approximately the same effect can be produced by providing multiple Hquid-access lines to the bed and periodically switching each stream to the adjacent line. Functionally, the adsorbent bed has no top or bottom and is equivalent to an aimular bed. Therefore, the four Hquid-access positions can be moved around the bed continually, always maintaining the same distance between the various streams. 

Most electrophilic substitutions in benzimidazole (31 R = H) occur primarily in the 5-position. In multiple bromination the order followed, 5 > 7 > 6,4 > 2, parallels molecular orbital calculations. In benzimidazole itself the 4(7)- and 5(6)-positions are tautomerically equivalent. Fusion of a benzene ring deactivates C-2 to electrophilic attack to such an extent that it is around 5000 times less reactive than the 2-position of imidazole. Strong electron donors at C-5 direct halogenation to the 4-position, whereas electron-withdrawing groups favor C-4 or C-6 substitution (84MI21). 

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