Point groups cubic

In cubic close-packing each molecule is surrounded by twelve others, whose interaction with the central molecule can be represented by a potential function of cubic point-group symmetry in case that the twelve molecules are spherically symmetrical or oriented at random. The energy change produced by this potential function,/say, is... 

Cubic point groups have four threefold axes (3 or 3) that mutually intersect at angles of 109.47°. They correspond to the four body diagonals of a cube (directions x+y+z, -x+y-z, -x-y+z and x-y-z, added vectorially). In the directions x, y, and z there are axes 4, 4 or 2, and there can be reflection planes perpendicular to them. In the six directions x+y, x-y, x+z,. .. twofold axes and reflection planes may be present. The sequence of the reference directions in the Hermann-Mauguin symbols is z, x+y+z, x+y. The occurrence of a 3 in the second position of the symbol (direction x+y+z) gives evidence of a cubic point group. See Fig. 3.8. 

Examples of three octahedron 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. 

What rests are the socalled cubic point groups T, Td, Th, O and Oh. The symmetry operators for these groups are conveniently described with reference to a cube, Fig. 1. 

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 ... 

This table can be used for all the other finite non-cubic point groups as well, as dbcussed in Sect. 8. Similar tables have been given by Griffith (4), but his signs are in some cases different from ours. 

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... 

Crystal energy matrix, 307 Crystal Hamiltonian, 294, 307 Crystals, organic molecular, 286, 327 Cube, Cn axes of, 19 commutation of, 23 symmetry operations of, 23 symmetry planes of, 20, 21 Cubic point groups, 66ff Current density, 109 Cyclobutadiene, v molecular orbitals of, 178-179... 

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,... 

Above 0c the unit cell of BaTiOs is cubic (point group m3m) with the ions arranged as shown in Figure 7.2. 

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 ... 

The degeneracies are given in parentheses. Multiplets of Kramers ions are at least twofold degenerate. Each CEF state can be given as a linear combination of free-ion J,M) states. For Ce (7= ) one has for the Kramers doublet and quartet (cubic point group),... 

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