Octahedral molecules point groups

This is the point group to which all regular octahedral molecules, such as SFe (Figure 4.12b) and [Fe(CN)6], belong. 

All regular fefrahedral molecules, which belong to fhe poinf group (Section 4.2.8), may show such a rofational spectrum. However, those spherical rotors that are regular octahedral molecules and that belong to the Of, point group (Section 4.2.9) do not show any such... 

In addition, G and F matrix elements have been tabulated (see Appendix VII in Nakamoto 1997) for many simple molecular structure types (including bent triatomic, pyramidal and planar tetratomic, tetrahedral and square-planar 5-atom, and octahedral 7-atom molecules) in block-diagonalized form. MUBFF G and F matrices for tetrahedral XY4 and octahedral XY molecules are reproduced in Table 1. Tabulated matrices greatly facilitate calculations, and can easily be applied to vibrational modeling of isotopically substituted molecules. Matrix elements change, however, if the symmetry of the substituted molecule is lowered by isotopic substitution, and the tabulated matrices will not work in these circumstances. For instance, C Cl4, and all share full XY4 tetrahedral symmetry (point group Tj), but... 

The most optimistic response to this situation is to claim that the force constant — t-bond order relationship is still valid, but that the reference points need to be changed V(CO)e itself is then a possible reference compound (76). The relationship can then only be quantified by using calculated orbital populations for the reference species, and can only be tested by more extended comparisons between calculated bond order and observed force constant. Precisely this test has been apphed to a whole group of substituted and unsubstituted octahedral carbonyls of groups VI and VII, the substituents in every case being hahde (77). The data used in fact were not force constants, but Cotton-Krainhanzel parameters this does not actually matter, since no reference molecules were used at all. Excellent agreement was found with an expression. 

A trigonal bipyramidal molecule belongs to point group It differs significantly from the tetrahedral AB4 and octahedral ABh in that all of the ligands, the B atoms, are not equivalent. If we use the numbering scheme... 

This problem of spurious or, as they are conventionally called, redundant coordinates always arises when there are sets of angles that form a closed group, as in the case just considered, in planar cyclic molecules and also in three dimensions (e.g., tetrahedral and octahedral molecules). Several of the examples discussed in Section 10.7 will illustrate the point further. Redundant coordinates can usually be recognized without much difficulty, though troublesome cases sometimes arise. 

Consider the molecule CIFjOj (with chlorine the central atom). How many isomers are possible Which is the most stable Assign point group designations to each of the isomers. 6.id The Structure for AliBr (Fig. 6. Ih) is assumed by both Al2Br6and ALCUin the gas phase. In the solid, however, the structures can best be described as closest packed arrays of halogen atoms (or ions) with aluminum atoms (or ions) in tetrahedral or octahedral holes. In solid aluminum bromide the aluminum atoms arc found in pairs in adjacent tetrahedral holes. In solid aluminum chloride, atoms are found in one-lhird of the octahedral holes... 

A molecule with four threefold axes and three proper fourfold axes is cubic or octahedral. If it has a centre of inversion the point group is Oh, otherwise it is O (very unusual). 

With respect to immobile chiroids, the appropriate symmetries are given by the familiar finite point groups C, (nonaxial), C (monoaxial), Dn (dihedral), T (tetrahedral), 0 (octahedral), and I (icosahedral). Molecules that belong to the first three groups are commonplace molecules with ground-state symmetries T [example tetrakis(trimethylsilyl)silane],39 O (example appoferritin 24-mer),40 and 1 (example human rhinovirus 60-mer),40 are relatively uncommon. 

Since our i-basis has even inversion symmetry, the matrix elements connected with the perturbation from a given water molecule are independent of whether this molecule is situated on one or on the other side of the central ion. This means that if we want to discuss the perturbation from our six water molecules with octahedrally positioned ligators (point group symmetry Tn), we can as well take into account only three of them, nos. 1, 2, and 3, say, and eventually multiply all perturbation matrix elements by two. One may say that the holohedrized symmetry (9, 21, 22, 23) of the three water molecules around the central ion is T1. ... 

Figure 6 do not represent the function (99) but merely join the calculated points. But the trend with the total number of electrons N shown in Figure 6 is important for comparison with the empirical results for this same group of tetrahedral and octahedral molecules, to which we now turn. 

Molecules with many symmetry operations may fit one of the high-symmetry cases of linear, tetrahedral, octahedral, or icosahedral symmetry with the characteristics described in Table 4-3. Molecules with very high symmetry are of two types, linear and polyhedral. Linear molecules having a center of inversion have symmetry those lacking an inversion center have Cx,v symmetry. The highly symmetric point groups Td,Oh, and 4 are described in Table 4-3. It is helpful to note the C axes of these molecules. Molecules with symmetry have only C3 and C2 axes those with Of, symmetry have C4 axes in addition to C3 and C2 and 4 molecules have C5, C3, and C2 axes. 

For an undistorted octahedral metal-oxide molecule MOg (Oj point-group symmetry), a treatment similar to that for the MO4 tetrahedron may be carried out using... 

An isolated n-atom molecule has 3n degrees of freedom and in—6 vibration degrees of freedom. The collective motions of atoms, moving with the same frequency and which in phase with all other atoms, give rise to normal modes of vibration. In principle, the determination of the form of normal modes for any molecule requires the solution of equation of motion appropriate to the n-symmetry. Methods of group theory are important in deriving the symmetry properties of the normal modes. With the aid of the character tables for point groups and the symmetry properties of the normal modes, the selection rules for Raman and IR activity can be derived. For a molecule with a center of symmetry, e.g. AXe, octahedral molecule, a non-Raman active mode is also IR active, whereas for the BX4 tetrahedral molecule, some modes are simultaneously IR and Raman active. 

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