Utilization of Symmetry Properties

The appearance of the same elements is evidently due to the equivalence of the two internal coordinates, Ar, and Ar2. Such symmetrically equivalent sets of internal coordinates are seen in many other molecules, such as those in Fig. 1-11. In these cases, it is possible to reduce the order of the F and G matrices (and hence the order of the secular equation resulting from them) by a coordinate transformation. 

If we choose a proper U matrix from symmetry consideration, it is possible to factor the original G and F matrices into smaller ones. This, in turn, reduces the order of the secular equation to be solved, thus facilitating their solution. Their new coordinates R are called symmetry coordinates. 

Here K is a symmetry operation, and the summation is made over all symmetry operations. Also, A i( ) the character of the representation to which R belongs. Called a generator, Ari is, by symmetry operation fC, transformed into K (Ari), which is another coordinate of the same symmetrically equivalent set. Finally, /V is a normalizing factor. 

As an example, consider a bent XY2 molecule in which Ar, and Ar, are equivalent. Using Ar, as a generator, we obtain 

The remaining internal coordinate, Aa, belongs to the At species. Thus the complete U matrix is written as 

the atoms surrounded by a bold line circle are those common to both coordinates. The symbols p and p denote the reciprocals of mass and bond distance, respectively. The spherical angle /o py in Fig. 1.24 is defined as 

The correspondence between the Decius formulas and the results obtained in Eq. 1.126 is evident. 

With the Decius formulas, the G-matrix elements of a pyramidal XY3 molecule have been calculated and are shown in Table 1.13. 

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In the case of dibromoethane, utilization of symmetry and other properties results in a reduction of the expression for the total potential energy to a dependence on only one variable which is the torsion angle. Because of this fortuitous circumstance it is convenient to present the conformation optimization calculations and results for that compound. 

The regular arrangement of polymer molecules in a crystalline region can be treated theoretically, utilizing the symmetry properties of the chain or crystal. With the advent of modem computers, the normal modes of vibrations of crystalline polymers can be calculated and compared with experiment. 

Hint Utilize the symmetry properties of the two-electron integrals. 

All of the other chapters in this book deal with the symmetries of finite (discrete) objects. We now turn to the symmetry properties of infinite arrays. The end use for the concepts to be developed here is in understanding the rules governing the structures of crystalline solids. While an individual crystal is obviously not infinite, the atoms, ions, or molecules within it arrange themselves as though they were part of an infinite array. Only at, or very close to, the surface is this not the case this surface effect does not, in practice, diminish the utility of the theory to be developed. 

Finally, in a novel application, some uniquely structured hexapyrrolidine derivatives of C60 with Th and D3 molecular symmetries have been synthesized and characterized by analytical methods and x-ray crystallography [260]. This work revealed strong luminescence, indicative of photophysical properties that are unusual in comparison with other fullerene derivatives. Therefore, the hexapyrrolidine adduct was utilized as a chromophore in the fabrication of a white light organic LED [261]. 

The availability of 18 in one synthetic step via direct oxidative functionalization of HCTD [19-21] allows exploration of this valuable compound s chemistry for the first time. Thus, the reaction sequence shown in Scheme 9 has been utilized to prepare a novel polycyclic alkene, 30, whose symmetry properties require that its central, tetrasubstituted C=C double bond be completely planar (in the isolated molecule). The constraints imposed by the polycarbocyclic cage framework cause the CCC bond angles about the central C=C double bond in 30 to deviate significantly from the preferred value of 120°, thereby introducing additional steric strain in this molecule beyond that which is associated with its framework alone. 

The symmetry of many molecules and especially of crystals is immediately obvious. Benzene has a six-fold symmetry axis and is planar, buckminsterfullerene (or just fullerene or footballene) contains 60 carbon atoms, regularly arranged in six- and five-membered rings with the same symmetry (point group //,) as that of the Platonic bodies pentagon dodecahedron and icosahedron (Fig. 2.7-1). Most crystals exhibit macroscopically visible symmetry axes and planes. In order to utilize the symmetry of molecules and crystals for vibrational spectroscopy, the symmetry properties have to be defined conveniently. 

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