Octahedral point group

For gaseous TcF the IR spectrum was measured in the range from 6 to 43 pm [70] and the Raman spectrum using laser excitation [71 ]. The spectra were interpreted in terms of the octahedral point group O/, [72 [. Considerable broadening of the and / 2g fundamentals was observed for TcF and ReFf,. This broadening was attributed to dynamic Jahn-Teller coupling. The values of fundamental frequencies for technetium and rhenium hexafluoride arc Riven in Table 10.6.A. 

Examination of the character table of the octahedral point group (Oj,) gives us the symmetry of the metal orbitals directly, as this atom is located at the origin of the c, y, z frame (Table 2.1). The last two columns indicate the symmetry of several functions of x, y, and z which have a direct link with the analytical expressions, in Cartesian coordinates, of these orbitals (Chapter 1, 1.4.1). 

It is also possible that symmetry-equivalent elements each give more than one operation to a class. For example, in the octahedral point group Oh there are four equivalent C3 axes, each of which contributes two operations which are in the same class (C3 and Cs ), and so the heading in the character table reads 8C3. One of these C3 axes is marked on the paper model of an octahedron from Appendix 4. 

The Couplitig-Coefficierits lJ ABC abc) for the Complex Form of a Doubly Degenerate Representation in the Octahedral Group, Following G. F. Koster et al.. Properties of tke Thirt i-Two Point Groups, MIT Press, MA, 1963, pp, 8, 52. 

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... 

For a spherical rotor belonging to the octahedral Of, point group, Table A.43 in Appendix A, in conjunction with the vibrational selection rules of Equation (6.56), show that the only allowed transitions are... 

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... 

In the trigonal point group 3, the axis of quantization is chosen along the three-fold axis, while the x and y axes may be selected anywhere in the plane perpendicular to the z axis. In the point group 3m, which occurs for many distorted octahedral complexes, there are also vertical mirror planes. The relation to the cubic axes is described by the transformation... 

In the point-group Td, p-orbitals transform in the same way as dxy, dyt, and dtx, owing to the absence of a center of symmetry. This has the important consequence that all six orbitals are mixed together and the electronic transitions have some d-p character they are therefore not forbidden and have higher intensities than the nearly pure d-d transitions of octahedral complexes. 

The hexaadducts [C60 M(PEt3)2 6] (M = Ni, Pd, Pt) involve an octahedral array of metal moieties in a similar fashion to C60 C(CO2Et)2 6 and possess the very rare point group Th (16). It is believed that each metal fragment binds to one fullerene C=C bond and sterically blocks the neighboring four, so that six moieties will block all 30 C=C bonds of C60. Hence, 7r-adducts involving more than six addends are likely to be difficult to prepare. 

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... 

We may use the set of five d wave functions as a basis for a representation of the point group of a particular environment and thus determine the manner in which the set of d orbitals is split by this environment. Let us choose an octahedral environment for our first illustration. In order to determine the representation for which the set of d wave functions forms a basis, we must first find the elements of the matrices which express the effect upon the set of wave functions of each of the symmetry operations in the group the characters of these matrices will then be the characters of the representation we are seeking. 

In a similar manner, we could determine the splitting of various sets of orbitals in environments of other symmetries which we may encounter in complexes, such as Td, DAtn D2d, and C2l., and indeed for any sort of symmetry we may encounter. An alternative and simpler way of obtaining this information is to use the results we have obtained for the octahedral case in conjunction with the correlation table given in Appendix IIB. In Table 9.3 are the results for a few point groups of particular interest. 

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