Site symmetries

A guest molecule may be endowed with a certain degree of symmetry and contain more than one equivalent quadrupolar nucleus, for example the three chlorine nuclei in chloroform (Symmetry C3v) or the four chlorine nuclei in carbon tetrachloride (Symmetry T ). When this molecule is a guest in an inclusion complex it may find itself in a site whose symmetry is less than that of the isolated molecule so that the 

Host Guest H G Ratio Equivalent Nuclei Resonance Frequencies Ref. 

---

Site symmetry Depth Probed Depth profiling Detection limits Lateral resolution Imaging/mapping... 

Tabic 6-2. Correlation diagram of the C2/, point group of the isolated T6 molecule (left column) with the C2i, factor group for solid T(, (right column) via the site symmetry C group (center). L, M, and N indicate the principal molecular transition dipole moments, while a, b, and c arc the crystalline axes. 

Site symmetry symbols, I, 128 Six-coordinate compounds stereochemistry, 1, 49-69 Six-membered rings metal complexes, 2, 79 Skeletal muscle sarcoplasmic reticulum calcium pump, 6, 565 Slags... 

It is also apparent that there remains much to be done in developing our understanding of the model interactions that best characterize the spectra of higher-valent Pu compounds and indeed of the actinides in general. The synthesis of new compounds with a variety of different site symmetries could be of particular value in developing more detailed crystal-field... 

The vibrations of the free molecule can be correlated with the vibrations of the crystal by group theoretical methods. Starting with the point group of the molecule Did)> the irreducible representations (the symmetry classes) have to be correlated with those of the site symmetry (C2) in the crystal and, as a second step, the representations of the site have to be correlated with those of the crystal factor group (D2h) [89, 90]. Since the C2 point group is not a direct subgroup of of the molecule and of D211 of the crystal, the correlation has to be carried out in successive steps, for example ... 

The spectra of S12 and Si2-CS2 clearly reflect the differing site symmetries of the S12 molecule The two Eg vibrations at 188 and 245 cm degenerate in Si2 CS2, split into their components in pure S12 with its non-degenerate site symmetry of C2h- The valence force constants calculated for S12 reproduce the observed wavenumbers within 14 cm [79]. The two most interesting constants are/r = 2.41 and/ r = 0.60 N cm ... 

F can he suppressed hy the high site symmetry of the central atom In many perovskite-like structures of the ABO3 type the lone pair of the B-cat-ion leads not to a structural distortion. In CsPbF3 under ambient conditions no lone-pair activity observed [27], but upon cooling a phase transition is observed that leads to less symmetrical surrounding of Pb by fluoride [28]. 

From the applications of Ni Mossbauer spectroscopy in solid-state research, it is clear that (1) information from isomer shift studies is generally not very reliable because of the smallness of the observed isomer shifts and the necessity of SOD shift corrections which turn out to be difficult, and (2) useful information about magnetic properties and site symmetry is obtained from spectra reflecting magnetic and/or quadrupolar interactions. 

If an atom is situated on a center of symmetry, on a rotation axis or on a reflection plane, then it occupies a special position. On execution of the corresponding symmetry operation, the atom is mapped onto itself. Any other site is a general position. A special position is connected with a specific site symmetry which is higher than 1. The site symmetry at a general position is always 1. 

Molecules or ions in crystals often occupy special positions. In that case the site symmetry may not be higher than the symmetry of the free molecule or ion. For example, an octahedral ion like SbClg can be placed on a site with symmetry 4 if its Sb atom and two trans Cl atoms are located on the fourfold axis a water molecule, however, cannot be placed on a fourfold axis. 

The Wyckoff symbol is a short designation it consists of a numeral followed by a letter, for example 8/. The cipher 8 states the multiplicity, that is, the number of symmetry-equivalent points in the unit cell. The / is an alphabetical label (a, b,c,...) according to the sequence of the listing of the positions a is always the position with the highest site symmetry. 

Every space group listed in the family tree corresponds to a structure. Since the space group symbol itself states only symmetry, and gives no information about the atomic positions, additional information concerning these is necessary for every member of the family tree (Wyckoff symbol, site symmetry, atomic coordinates). The value of information of a tree is rather restricted without these data. In simple cases the data can be included in the family tree in more complicated cases an additional table is convenient. The following examples show how specifications can be made for the site occupations. Because they are more informative, it is advisable to label the space groups with their full Hermann-Mauguin symbols. 

Inspection of the atomic positions reveals how the symmetry is being reduced step by step. In the aristotype usually all atoms are situated in special positions, i.e. they have positions on certain symmetry elements, fixed values of the coordinates, and specific site symmetries. From group to subgroup at least one of the following changes occurs for each atomic position ... 

The site symmetry is reduced. Simultaneously, individual values of the coordinates x, y, z may become independent, i.e. the atom can shift away from the fixed values of a special position. 

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. 

The symmetry reduction to the mentioned hettotypes of diamond is necessary to allow the substitution of the C atoms by atoms of different elements. No splitting of Wyckoff positions, but a reduction of site symmetries in necessary to account for distortions of a structure. Let us consider once more MnP as a distorted variant of the nickel arsenide type (Fig. 17.5, p. 197). Fig. 18.4 shows the relations together with images of the structures. 

Inspection of Equation 1.23 and consideration of the properties of 3-y and 6-j symbols confirm that only even A--values contribute to crystal field splitting. Further, it indicates that mixing between levels belonging to different / multiplets can only occur if terms with k [Pg.14]

Taking into account that Bq parameters represent the coefficient of an operator related to the spherical harmonic ykq then the ranges of k and q are limited to a maximum of 27 parameters (26 independent) Bq with k = 2,4,6 and q = 0,1,. .., k. The B°k values are real and the rest are complex. Due to the invariance of the CF Hamiltonian under the operations of the symmetry groups, the number of parameters is also limited by the point symmetry of the lanthanide site. Notice that for some groups, the number of parameters will depend on the choice of axes. In Table 2.1, the effect of site symmetry is illustrated for some common ion site symmetries. 

Mj = Mjmax-this is the only case, which can, in principle, lead to a singledeterminant wave function. This situation may be encountered in some limited highly axial systems, such as Dy-0+ [37]. However, usual compounds do not have any site symmetry, which means that several projections of the /-manifold will be mixed by the low-symmetry components of the CF, leading to a multideterminantal character of the wave function. 

---