Basic Point Group Theory

Let us be clear that we are trying to give a basic introduction to the use point group theory rather than a mathematical seminar. This is a low level how to do it introduction  

In the study of crystals there is a similar system of mathematics for what is called space group theory, but here we are concerned with the symmetry properties of individual molecules relative to some point within the molecule. This body of mathematics is called point group theory.  

FIGURE 18.1 Flowchart for selection of the character table appropriate for a molecule. (From Harris, D.C. and Bertolucci, M.D., Symmetry and Spectroscopy, An Introduction to Vibrational and Electronic Spectroscopy, Oxford University Press, New York, 1978. With permission.) 

FIGURE 18.2 The Cartesian arrow representation of the atoms in the water molecule. Note that we have maintained a right-handed coordinate system with the positive y-axis pointing to the back, but what really counts is that the y-coordinate changes sign when the image is reflected in the plane of the drawing. 

Once you can identify the specific symmetry operations a given molecule possesses you can then follow the flowchart to identify the proper name of the point group, here it is Cay. For this point group, there are four symmetry operations and we can state the mathematical definition of a point group while you visualize the operations in your mind. Chemists usually have developed minds-eye visualization due to studying organic chemistry and point group symmetry operations should be easy for chemists. 

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Counting the isomers arising by addition to, or substitution in, a basic framework is a mathematical problem with many practical applications in chemistry. In classical organic chemistry, for example, the number of derivatives of a compound was often cited as proof or disproof of structure. Point group theory that uses concepts familiar to most chemists and is easy to apply when the number of addends/substituents is small provides a unified method for deciding, for example, the number of dihydrides C70H2 of fullerene C70, or the number of trihalo-derivatives C2oHi7FClBr of dodecahedrane. All that is needed to determine such matters is the availability of the permutation character. Ter, of the atoms in the parent molecule. 

Since the molecular symmetry is important in vibrational spectroscopy few explanations will be given shortly with regards to comments within the tables. For more details the reader is referred to basic literatures on the theory of molecular symmetry. Vibrations of functional groups of a polymer chain can be described by the so-called point groups. The point group theory means that during the vibration at least one point in the molecule remains unaffected. Furthermore, for molecular units with some symmetry, point group theory can be used to predict how many vibrational modes should appear in the IR and Raman spectra. As different molecular symmetries lead to different numbers of IR and Raman active bands this is a... 

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