Cubic group

Figure 11.9 Arrangementt of ions in silicate garnet (grossularite). Showing tetrahedral, octahedral, and quasi-cubic groups. After Geller (1967).

We determine whether the molecule belongs to one of the special groups, that is, C, D / or one of those with multiple high-order axes. Only linear molecules can belong to or D h, so these cannot possibly involve any uncertainty. The specially high symmetry of the others is usually obvious. All of the cubic groups, T, Th, Trf, O, and 0/M require four C3 axes, while /... 

Each of these configurations, except d1 and d9, will give rise to terms under the action of HER as a perturbation. The terms, of course, bear the labels of the cubic group, here Oh. The terms which arise are determined qualitatively by the decomposition of the group theoretical direct product of the electrons (or holes) involved into irreducible representations of the cubic groups. A t2s or eR orbital set more than half-full is treated as the equivalent number of holes, and a filled one is ignored. Then, for instance... 

We shall examine the behaviour of these functions under various symmetry operations. We need only consider the behaviour of functions centred at the origin, as functions centred elsewhere display the same behaviour together with a possible translation of the centre that is easily determined. Cubic groups are excluded as the spherical harmonics are less well suited as basis functions for these cases. 

The values of 2% If n (where n is the order of the axis of rotation) that satisfy eq. (16) and therefore are compatible with translational symmetry, are shown in Table 16.1. It follows that the point groups compatible with translational symmetry are limited to the twenty-seven axial groups with n= 1, 2, 3, 4, or 6 and the five cubic groups, giving thirty-two... 

The spin-orbit does not interact for the A-term or E-term manifolds as these kets do not involve the angular momentum (in the cubic groups). Consequently all the g-limes degenerate energy levels within the model space possess zero matrix elements of the spin-orbit operator. 

The ground T-terms of the cubic groups are poorly described by a singleterm function. The Tig-terms arising from the different free-atom F- and P-terms possess the configuration interaction (Cl) because of the nonzero matrix element... 

When the symmetry of the complex is lower than that of the cubic group, a low-symmetry CF potential needs to be added to the SH ... 

The kets appropriate for such a Hamiltonian belong to the Ti- or T2-terms of the cubic groups. These are listed in Table 12. 

Table 64 Character tables of some cubic groups and their double groups...

The application of the reduction formula is exemplified by the decomposition of the D2 representation of the group R3 (Table 65) in terms of the irreducible representations of the cubic group O having the character table according to Table 64. Now the appearances of the individual irreducible representations are evaluated according to the reduction formula of the form... 

The body-centered cubic Group VA metals dissolve a considerable amount of hydrogen before distorting to either a b.c. tetragonal or orthorhombic structure. These behave somewhat more like solid solutions of hydrogen in metal than do the more definite hydrides previously discussed. They appear to have the stoichiometric formula MH, but this has never been reported, the maximum being about... 

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