Tetrahedral point group

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

The methane molecule is a very important molecule in organic chemistry, the geometry around the tetravalent carbon atom being basic to the understanding of the structure, isomerism and optical activity of a very large number of compounds. It is a tetrahedral molecule belonging to the tetrahedral point group, Td. 

As a first illustration let us consider the optical transitions in a tetrahedral complex of Co(II). The ground state belongs to the A2 representation of the tetrahedral point group Td, and there are two excited states of F, symmetry and one of T2 symmetry. The character table for Td tells us that the coordinates x, y, and z form a basis for the T2 representation. For the A2 F, transitions we then see that the intensity integral will span the representations in the direct product of A2 x F, x F2, and this reduces as follows ... 

For simplicity and brevity, we consider the pure rotational subgroup T of the tetrahedral point group T4. The extension of the analysis to Td is straightforward. We want to And the complete set of symmetry operators Z for which... 

In the tetrahedral point group d, cL, d,y) and (p py, Pz) both transform according to the tz irreducible representation and generate three d-p hybrids which point towards the vertices of a cube not utilized by the tetrahedron (see Fig. 11). 

For groups with equivalent sets of operations the corresponding values of gc will be greater than 1. For example, in the tetrahedral point group, T, the character table in Appendix 12 states that there is one operation in the identity class, 8 operations in the C3 class, 3 in the C2 and so on. If we sum the number of operations in aU classes we obtain the order of the group, i.e. ... 

This is the point group to which all regular tetrahedral molecules, such as methane (Figure 4.12a), silane (SiFl4) and nickel tetracarbonyl (Ni(CO)4), belong. 

As we proceed to molecules of higher symmetry the vibrational selection rules become more restrictive. A glance at the character table for the point group (Table A.41 in Appendix A) together with Equation (6.56) shows that, for regular tetrahedral molecules such as CH4, the only type of allowed infrared vibrational transition is... 

Collectively, the symmetry elements present in a regular tetrahedral molecule consist of three S4 axes, four C3 axes, three C2 axes (coincident with the S4 axes), and six mirror planes. These symmetry elements define a point group known by the special symbol Td. 

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 chromate anion is a highly soluble, toxic tetrahedral complex (point group Tj) that occurs in oxidized, neutral-basic solutions. It is also one of a small number of aqueous complexes that have been thoroughly characterized by spectroscopic measurements on numerous isotopic compositions (Muller and Kbniger 1974), so it will be possible to check the vibrational model against real data. Here the MUBFF is applied under the assumption that aqueous chromate can be approximately modeled as a gas-phase molecule. 

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