Most of us become aware of symmetry very early in life and we usually associate it with pleasing proportions, beauty or harmony. Natural forms including starfish, fir trees, Fujiyama, snow flakes and man-made objects including most houses and cars, indeed almost all objects we come in daily contact with possess a symmetry that we implicitly recognize. To a first approximation the human body itself has a left-right symmetry.

In this part we will describe the four (or really five) symmetry operations which are of importance in the world of human dimensions as well as at the molecular level. To do this we will utilize a portion of that body of mathematical knowledge known as "group theory." Although chemical applications of group theory on a major scale dates only from the 1950's the mathematical background goes back to 1832. At that time a certain French mathematician, Galois, was to fight a duel over an affair with a lady friend (although it may also have had some political implications). The night before the duel Galois, in a long letter to a friend, formulated group theory and laid the basis for further development. He did not participate in the later developments because he lost the duel. We can follow some of his work below.

A group consists of a collection of elements that obey the four group postulates:

1. The products of any two elements in a group are also an element in the group. If A is in G and B is in G, then AB and BA are also in G.

2. In every group there is the identity element, I, such that AI = IA = A for all A in G.

3. The associative law must hold. If A, B and C are all in G then A(BC) = (AB)C.

4. Every element in the group has a reciprocal that is also in the group. For every A in G there exists an element R in G such that AR = RA = I

Some groups have the property that all elements in the group commute. That is AB = BA for all A and B in G. Such groups are called Abelian groups.

There are two major types of groups of interest to chemists - point groups and space groups. The former are usually used to analyze molecular symmetry while the latter are usually used to analyze crystallographic symmetry. There are some 50 point groups of interest to chemists, but there is no limit in principal. On the other hand, there are 232 space groups in three-dimensional space and all crystals must belong to one of these.

In this course we shall be primarily concerned with point groups and their application in classifying molecular properties. Point groups of interest are those that involve molecular symmetry operations that leave at least one point in the molecule in a fixed position. Of course, if the molecule is in a fixed position then the identity operation (E) can always be applied and it will remain in the same position. In other words, the operation, E, can always be applied to any molecule. There are four non-trivial operations which can transform the molecules in such a way that the numbering of equivalent atoms may change after the symmetry operation has been carried out but the geometry remains identical (except for numbering) to the original. The four operations and their conventional symbols are:

1. Rotation about an axis through an angle equal to 360/n. This is a Cn operation. Examples include a C2 axis in H20, 3 C2 axes and a C3 axis in BF3 and a C6 axis and 6 C2 axes in benzene.

2. Reflection in a plane - 6. Examples are the two planes in H20. The three vertical planes (6v) and one horizontal plane (6n) in BF3 and the six vertical planes and one horizontal plane in benzene. By convention, the plane of reflection being perpendicular to the highest rotation axis (the one with the largest "n") is called the horizontal plane, while the planes which include the highest rotation axis are called the vertical planes. These may be grouped into categories that reflect similarities.

For example, the three planes of symmetry which bisect the bonds in benzene are in one category while those bisecting the angles are in another. The symbol 6 is related to the German word "spiegel" (mirror).

There may be many mirror planes in a molecule.

3. A third symmetry operation is inversion in the center - i. A molecule may either have an inversion operation or not. For example, H20 and BF3 cannot be inverted while benzene has a center of symmetry.

4. The fourth symmetry operation is the improper rotation - Sn. This consists of a rotation about an axis by an angle 360/n followed by reflection in the plane perpendicular to the axis of rotation. Methane possesses four S4 axes. Each of these bisects a HCH bond. BF3 possesses an S3 axis that is equivalent to its C3 axis. In this case it appears that nothing new has been added. However, consider the molecule, PF5. Here the C3 axis and S3 axis are again coincident but the result of performing the symmetry operation on the latter are resulting in an interchange of the axial fluorines.

The collection of all symmetry operations that can be performed on a molecule form a group and, when these symmetry operations are property formulated, they can be shown to obey the Galois group postulates. Of course, many molecules may belong to the same group.

When the point group of a molecule has been identified, it is often possible to discuss many of its physical and chemical properties in terms of specific group theoretical symbols (called "irreducible representations") which are associated with that point group. An entire course could be devoted to this topic alone. It is our purpose here to develop a useful working knowledge of some important chemical applications of group theory so that a student can read the current literature with both confidence and competence.

As it turns out, symmetry operations can be represented as matrices. So the collection of all symmetry operations which can be performed on a particular molecule or molecular wavefunction can be formulated as a set of symmetry operations which will, in mathematical form, be a collection of matrices. As it turns out, it is only rarely necessary to know the full matrix. In fact, the item of interest is the sum of the diagonal elements of the matrices and this sum is called the "character" (or trace or spur) of the matrix. Fortunately, there are a few simple rules that can be used to calculate the "character" associated with a molecular property once the proper point group of the molecule has been identified. Furthermore, this "character" will be associated with a representation of the molecule and it will be possible in a simple way to break up (or "decompose") any representation into a set of one or more of the irreducible representations of the point group. This is usually done with the help of a set of "character tables" which summarize the important information about the matrix representation of the group and some of the transformation properties of useful mathematical functions. The character tables for the most useful point groups are listed in the appendix. Let us examine a typical point group.

In the upper left-hand corner the name of the group is listed. As an example let us consider the molecule ammonia, NH3, which belongs to the point group C 3v. The character table for C3v is:

C3v* E 2C3 36v

Al 1 1 1 Z X2+Y2 Z2

A2 1 1 -1 RZ

E 2 -1 0 (X,Y)(Rx,Ry)(X2-Y2)(XZ,YZ)

The operations in this group include the identity operation, 2 rotational operations and three reflections. As it turns out the rotation by 120° and the rotation by 240° have similar matrix forms (they belong to the same "class" - a matter to be discussed below) and they can be listed together. Similarly the three reflections also belong to the same class and are also listed together.

In the point group of water, H20, which is in C2v there are two reflections but they belong to different classes. The character table for C2v is:

C2v E C2 6v(XZ 6v(YZ)

Al 1 1 1 1 Z

A2 1 1 -1 -1 RZ

B1 1 -1 1 -1 X, Ry

B2 1 -1 -1 1 Y, Rx

A simple way of determining if two operations (A and B) belong to the same class is to see if there is another operation (C) in the group which will take A into B or B into A. In the point group C3v a rotation by 120° (C) will take one of the reflections into another. There is, of course, a formal mathematical way of formulating membership in a class but this simple method provides a pictorial method. Actually the character tables already classify the appropriate information and the usual problem is to verify that the molecule of interest fits the character table, not to see if the character table is correctly formulated.

Now it turns out that the number of classes in a group is equal to the number of irreducible representations. So there are four irreducible representations in C2v and 3 in C3v. In C2v each of these is one dimensional and such representations are given the symbol A or B (according to whether the character under the major rotational axis is +1 or –1 respectively). In C3v there are two one-dimensional irreducible representations and one two-dimensional irreducible representation. For three-dimensional irreducible representations the symbol is usually T (although occasionally F).

A simple way of interpreting the characters of the one-dimensional irreducible representation and associating them with functions can be seen by a few simple examples. In both C2v and C3v a vector pointing along Z will remain unchanged under all the symmetry operations of the group and this is reflected in the +1 character under all the operations. The vector along Z is then said to "belong" to the irreducible representation Al in C2v and C3v. Note that rotational motion about the Z axis (say clockwise) is unchanged by a rotation by 180° in C2v about the C2 axis or by 120° in C3v about the C3 axis, thus the +l character. But reflection in a plane that includes the major axis (e.g. 6v in C3V) changes the sense of rotation (clockwise to counterclockwise) and the result is the negative sign of the character. So it turns out .in both of these groups Rz belongs to the irreducible representation A2.

Simple functions belong to one-dimensional representation but in the group C3v the pair X and Y belong to the two-dimensional representation E. This has the interpretation that in this group the X and Y axis can be placed in any position perpendicular to Z and they must always be treated as a pair. Put another way, the xy plane is well defined in the molecule but the location of the pair of perpendicular axes is completely undefined.

There are many ways of interpreting the meaning of "Z belongs to Al in C3v". It could mean that the orbital pz, which has a positive end and a negative end, belongs to Al. It could also mean that an electric field of a light beam polarized along the z-axis belongs to Al. It could also mean that translational motion in the z direction belongs to Al. These, and other interpretations of a single mathematical statement are what makes group theory, or more properly, symmetry concepts, so important in chemistry.

The proper use of these powerful tools is not difficult to develop and, with only a few examples, relatively easy to follow.

In this part of the course we expect that you will develop the ability to classify a molecule into an appropriate point group (assuming you have a set of character tables available to you) and to follow a few simple examples. In later chapters we will utilize group theory and especially group theoretical notation in the discussions of structure, bonding and spectroscopy.