The Cubic Point Groups

Note that dj2 is the short notation for d7l2 2 2, as it appears in the cubic point groups (Appendix VII). Similarly, fti, fxtj and are the abbreviated symbols for... 

The following operators were used in Chapter 7 as representative operators of the five classes of the cubic point group O E, R(2n/3 [1 1 1]), R(n/2 z), R(n z), R(n [1 1 0]). Derive the standard representation for these operators and show that this representation is irreducible. [Hint You may check your results by referring to the tables given by Altmann and Herzig (1994) or Onadera and Okasaki (1966).]... 

Molecules belonging to the cubic point groups can, in some sense, be fitted symmetrically inside a cube. The commonest of these are Td, Oh and I they will be simply exemplified ... 

The electronic structure, in particular the electronic spectroscopic properties, of the whole class of cobalt amine complexes may be reduced to a discussion of the central Co-Ne core. This disregards, of course, the charge-transfer transition that in these compexes typically occurs around 250 nm. The geometrical structure is either octahedral or is defined in terms of a subgroup of the cubic point group Oh, where the... 

Extended Htickel theory calculations are the maximum level of approximation that can be employed for the calculation of the eigenvalue spectra for the regular orbit cages of the cubic point groups, since different bond lengths have to be accommodated on the His orbital... 

Attention should perhaps be drawn to the characteristic symmetry of the cubic system which is not, as might be supposed, the 4-fold (or 2-fold) axes of symmetry or planes of symmetry but four 3-fold axes parallel to the body-diagonals of the cubic unit cell. This combination of inclined 3-fold axes introduces either three 2-fold or three 4-fold axes which are mutually perpendicular and parallel to the cubic axes. Further axes and planes of symmetry may be present but are not essential to cubic symmetry and do not occur in all the cubic point groups or space groups. 

Precisely the same method as for D4 can be used to construct generalized eigenvalue tables for all, except the cubic point groups. The generalized commutation relation, Eq. (7), holds for any n... 

The inversion operation commutes with all rotations and reflections and so simply leads to an extra eigenvalue. The operator 6h also commutes with both Cn and Q and so again leads to an extra eigenvalue. For all, except the cubic point groups, the construction of the eigenvalue table is then straightforward. Results for all the point groups are listed in Appendix A, in a way similar to Table 2 for D4. 

As said the situation is a little more complex for the cubic point groups. We start with the point group T, which may be generated from the operators C2, Q and C3. The first two operators commute, and the generalized commutation relations with respect to C3 are easily seen to be ... 

The CH4 molecule has symmetry. The relationship between a tetrahedron and cube that we illustrated in Figure 4.6 is seen formally by the fact that the point group belongs to the cubic point group family. This family includes the and Oj, point groups. Table 4.3 shows part... 

Crick and Watson were the first to suggest that small viruses were built up of small protein subunits packed together symmetrically to form a protective shell for the nucleic acid. They reasoned that formation of small identical molecules was an efficient use of the limited information contained in the virus nucleic acid. They also realized that, of the types of symmetry possible for a three-dimensional structure enclosing space, only the cubic point groups could lead to an isometric particle, which was the known symmetry of many viruses at the time. Three types of cubic symmetry exist namely,... 

The CH4 molecule has Tj symmetry. The relationship between a tetrahedron and cube that we illustrated in Figure 5.6 is seen formally by the fact that the point group belongs to the cubic point group family. This family includes the Tj and O], point groups. Table 5.3 shows part of the character table. The C3 axes in CH4 coincide with the C—H bonds, and the C2 and 54 axes coincide with the x, y and z axes defined in Figure 5.6. Under symmetry, the orbitals of the C atom in CH4 (Figure 5.20a) are classified as follows ... 

Contributions to the magnetoelastic interaction that contain the rotational tensor are important only in an external magnetic field. This will be considered separately in sect. 2.6. For the moment we consider only pure strain contributions. In cubic crystals the components of e can be grouped to form irreducible representations of the cubic point group (Thalmeier and Fulde 1975, du Tremolet de Lacheisserie et al. 1978) ... 

In the cubic point groups (T, T, Tj, O, O ), the doubly degenerate modes have only diagonal tensor elements related by a threefold axis of rotation, so that we have... 

The SFg molecule, 5.5, belongs to the point group, which is one of the cubic point groups. The relationship between the octahedron and cube is shown in Fig. 5.26a the X, y and z axes for the octahedron are defined as being parallel to the edges of the cube. In an octahedral molecule such as SF5, this means that the x, y and z axes coincide with the S F bonds. Table 5.4 gives part of the character table, and the positions of the rotation axes are shown in Fig. 5.26b. The SFg molecule is centrosymmetric, the S... 

The cubic point groups are point groups containing symmetry elements for proper rotation by less than 180° around more than one axis. 

FIGURE 13.13 The five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Collectively, these five shapes represent the cubic point groups. 

Let us now consider the necessary conditions for the appearance of phonons in impurity-ion electronic spectra. The presence of a substitutional defect in an otherwise perfect crystal removes the translational symmetry of the system and reduces the symmetry group of the system from the crystal space group to the point group of the lattice site. Loudon [26] has provided a table for the reduction of the space group representations of a face-centered cubic lattice into a sum of cubic point-group representations. A portion of that table is shown in Table 1 here. Consider an impurity ion that undergoes a vibronic electric-dipole allowed transition, with T and Tf the irreducible representations of the initial and final electronic states. Since the electric dipole operator transforms as Tj in the cubic point group, Oh, the selection rule for participation of a phonon is that one of its site symmetry irreducible representations is contained in the direct product T x Ti X Tf. 

Another arrangement is provided by the cubic symmetry the cubic point groups. This kind of arrangement does not exist for smaller proteins. Only oligomeric proteins with 12n protomers may possess cubic symmetry. This symmetry designated as T (or 23) is characterized by more than one rotational axis greater than twofold. For example, the tetrahedron possesses four threefold axes. The threefold symmetry as previously mentioned is not possible when asymmetric protomers of a protein are at the vertices of a tetrahedron. However, it can be obtained if a group of three subunits are located at the four vertices of a tetrahedron. This involves a number of 12 asymmetric protomers. 

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