Character Tables 13

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The symmetry of molecules can be treated systematically using group theory. A motion about a point, line, or plane that leaves a molecule looking the same is called a symmetry operation. The point, line, or plane about which the motion occurs is a symmetry element of the molecule

An example of a symmetry operation is the rotation by 120o around the C3 axis in CH3Cl. This rotation leaves the molecule apparently unchanged. One of the symmetry elements of CH3Cl is a C3 axis.

The set of symmetry elements possessed by a particular molecule determines the point group to which the molecule belongs.



Associated with each point group is a character table. As an example, the C3v character table is shown at the right. The symmetry elements in the C3v point group are along the top. If two or more symmetry axes are the same as each other, except in the reverse direction, they are listed together (2C3). If two or more planes are similar to each other, but in different positions on the molecule, they are listed together.



The symbols in the left column (A1, A2, E) are irreducible representations of the group. A full discussion of irreducible representations involves matrices13 and is more than we want (or need) to get into here. It is convenient to think of the symbols in the left column as representing symmetry types14. The symmetry types are used to label the symmetries of various molecular vibrations or molecular orbitals that are associated with a molecule belonging to this point group. For example, the totally symmetric vibration shown at the right is of symmetry type A1.



The numbers under the symmetry elements are characters. The only thing that we will do with characters is to notice that the totally symmetric symmetry type has all characters = 1.

The letters at the right of the table, z or xy for instance, represent the coordinates or products of the coordinates. It should be noted that (x,y) is NOT a product of x and y. (x,y) represents a two-dimensional combination of the two coordinates.

The coordinates can be used to represent atomic orbitals (z = pz) or to identify IR- and Raman-active transitions.



Partial support for this work was provided by the National Science Foundation's Division of Undergraduate Education through grant DUE #9751605 and by CAChe Scientific through a Higher Education Program grant.

The PCOL community that partial support for this work was provided by the National Science Foundation's Division of Undergraduate Education through grant DUE #9950809. Additional support was provided by the Camille and Henry Dreyfus Foundation. PCOL faculty also acknowledge the National Science Teachers Association which awarded the PCOL Faculty Consortium the 1998 Gustav Ohaus Award for Innovation in College Science Teaching.

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