Examples and Applications of Symmetry

As well as departure from octahedral symmetry of the tetragonal type discussed above, angular distortions in which M—L bond lengths are preserved unchanged present a fairly obvious case for the application of the AOM. An octahedron squashed or elongated in the C3 axis is an example. In the resultant D3d symmetry the -orbitals are split into three sets, one of symmetry alg and two of symmetry eg. In fact, the expressions for and (p = alg or eg) in terms of the angle of distortion are not simple and the existence of two sets of the same symmetry label creates complications. However, it is readily shown that it is a two-parameter problem in the AOM, eG and en, the same level as for LFT (see equation 9). In principle, the two energy separations available should allow the evaluation of both parameters. 

In the next chapter, we will present various chemical applications of group theory, including molecular orbital and hybridization theories, spectroscopic selection rules, and molecular vibrations. Before proceeding to these topics, we first need to introduce the character tables of symmetry groups. It should be emphasized that the following treatment is in no way mathematically rigorous. Rather, the presentation is example- and application-oriented. 

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

More advanced applications of symmetry (not discussed here) involve the behaviour of molecular wavefunctions under symmetry operations. For example in a molecule with a centre of inversion (such as a homonuclear diatomic, see Topic C4). molecular orbitals are classified as u or g (from the German, ungerade and gerade) according to whether or not they change sign under inversion. In... 

Molecular symmetry and ways of specifying it with mathematical precision are important for several reasons. The most basic reason is that all molecular wave functions—those governing electron distribution as well as those for vibrations, nmr spectra, etc.—must conform, rigorously, to certain requirements based on the symmetry of the equilibrium nuclear framework of the molecule. When the symmetry is high these restrictions can be very severe. Thus, from a knowledge of symmetry alone it is often possible to reach useful qualitative conclusions about molecular electronic structure and to draw inferences from spectra as to molecular structures. The qualitative application of symmetry restrictions is most impressively illustrated by the crystal-field and ligand-field theories of the electronic structures of transition-metal complexes, as described in Chapter 20, and by numerous examples of the use of infrared and Raman spectra to deduce molecular symmetry. Illustrations of the latter occur throughout the book, but particularly with respect to some metal carbonyl compounds in Chapter 22. 

Symmetry properties can be used both in the direct and in the reciprocal space, for example, to form matrices in direct space, such as F and or to diagonalize F(k) more efficiently. The application of symmetry to direct space... 

The first step in the application of symmetry to molecular properties is therefore to recognize and organize all of the symmetry elements that the molecule possesses. A symmetry element is an imaginary point, line, or plane in the molecule about which a symmetry operation is performed. An operator is a symbol that tells you to do something to whatever follows it. Thus, for example, the Hamiltonian operator is the sum of the partial differential equations relating to the kinetic and... 

We will return to the application of symmetry to wavefunctions in a later section. Symmetry in molecules has several immediate consequences. For example, the dipole moment of a molecule depends in part on how the atoms in the molecule are arranged. It can be shown that any molecule whose structure has a point group symmetry of C, C , or C , with n > 1, is polar, and molecules that do not have such symmetry are nonpolar. Further, it can also be shown that a three-dimensional molecule that does not have an S axis is chiral. An S axis may not be explicitly listed in the point group, so care must be used with this requirement. Chirality is an important issue in organic chemistry and is the basis of stereochemistry. 

In Section 4.8 we found how the matrix representation of a set of basis vector transformations for a point group can sometimes be made simpler. For example, the 2 x 2 matrix representation for x, y at O in H2O can be reduced to two T x 1 matrices (i.e. a simple number for each operation) for x and y. This process of simplifying a representation to a set of irreducible standard representations is central to the application of symmetry in chemistry and corresponds to finding the fundamental modes of vibration that underlie molecular motion. 

Type II restriction enzymes have received widespread application in the cloning and sequencing of DNA molecules. Their hydrolytic activity is not ATP-depen-dent, and they do not modify DNA by methylation or other means. Most importantly, they cut DNA within or near particular nucleotide sequences that they specifically recognize. These recognition sequences are typically four or six nucleotides in length and have a twofold axis of symmetry. For example, E. coU has a restriction enzyme, coRI, that recognizes the hexanucleotide sequence GAATTC ... 

The most characteristic feature of any crystal is its symmetry. It not only serves to describe important aspects of a structure, but is also related to essential properties of a solid. For example, quartz crystals could not exhibit the piezoelectric effect if quartz did not have the appropriate symmetry this effect is the basis for the application of quartz in watches and electronic devices. Knowledge of the crystal symmetry is also of fundamental importance in crystal stmcture analysis. 

---