Group symmetry coordinates

By die transformation (3.3) a substantial step is made in die transition from experimental intensities into quantities characterizing molecular structure. At die first place, a natural separation between dipole derivatives associated with bond stretchings and angle deformations is achieved. In some cases the dp/dSj derivatives can be associated with vibrations localized within certain atomic groupings. Such distortions may be described by local group symmetry coordinates. Snyder [27] first applied dipole moment derivatives with respect to group symmetry coordinates as basic parameters in infrared intensity analysis on a series of crystalline n-alkanes. The procedure described in his work will be discussed later in this section. 

The index G refers to a molecular-fixed Cartesian system in which die dipole moment derivatives with respect to normal coordinates are obtained. The summation is over the group symmetry coordinates. Eq. (3.70) can be solved if the Cartesian ctmqionents of dpo/dQ are known. Introducing Eq. (3.70) into Eq. (1.50) the following relation between the dipole derivatives dpQ/dSj and the intensity A)( is obtained... 

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

During the investigation of the structure of brookite, the orthorhombic form of titanium dioxide, another method of predicting a possible structure for ionic compounds was developed. This method, which is described in detail in Section III of this paper, depends on the assumption of a coordination structure. It leads to a number of possible simple structures, for each of which the size of the unit of structure, the space-group symmetry, and the positions of all ions are fixed. In some cases, but not all, these structures correspond to closepacking of the large ions when they do, the method further indicates... 

In the following problems the positions of symmetrically equivalent atoms (due to space group symmetry) may have to be considered they are given as coordinate triplets to be calculated from the generating position x,y,z. To obtain positions of adjacent (bonded) atoms, some atomic positions may have to be shifted to a neighboring unit cell. 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates. For example, denoting in Fig. 6.1 by sx, s2, s3,, v4, j5, s6 the stretching coordinates of the six C-H bonds, the internal symmetry coordinates are linear combinations... 

We now allow nuclear motion and seek vibrational wave functions corresponding to states i i and tjfg. We assume throughout that the subunits have the same point group symmetry in both oxidation states (M and N), and then it is only necessary to consider explicitly totally symmetric normal coordinates of the two subunits (4, 5). Let us assume that there are two on each... 

The two BH3 groups are coordinated to two adjacent phosphorus atoms of the P5 ring in cyclo-1,2-(BH3)2(cyclo-P5Ph5) (Figure 11.7). Only one dia-stereomer is present in the unit cell, corresponding to two enantiomers related by the crystallographic centre of symmetry. In solution, however, a complex mixture of isomers is evident from P NMR spectra. 

The oxyanions as ligands may be classified according to (a) the structural type of the oxyanion (X02, X03, X04 or X06) (b) the coordination number of the oxyanion (1-18), i.e. the number of metal atoms to which a single oxyanion may be coordinated (c) the mode of coordination of the oxyanion, i.e. monodentate, bidentate, tridentate, etc. and (d) the number of oxyanions per metal atom, the stoichiometry p, from one to six, i.e. [M(XO ) ]. Table 1 lists the oxyanions that will be considered in this section according to their structural types, with their approximate stereochemistry and point group symmetry. The carbon-containing oxyanions will be described in Chapter 15.6, and the cyanates in Chapter 13.5, For reasons of space this review will be primarily restricted to mononuclear oxyanions. Figures 2-5 illustrate the mode of coordination of the oxyanions as a function of their coordination number 1-18. 

---