Introduction to Symmetry and Point Groups

Symmetry is especially important in carrying out molecular orbital calculations.

Although SPARTAN and GAUSSIAN generally take care of the detailed symmetry manipulations for us, it is important that we

Useful references are:

Symmetry elements are geometric entities that are used to minipulate molecules so as to transform them from one spatial orientation into another, indistinguishable, orientation. For the analysis of the large majority of organic molecules, only two symmetry elements are needed:

These elements can be defined as follows:

For example, chloroform has three symmetry planes, defined by the C-H bond and each C-Cl bond, and bisecting the opposite Cl-C-Cl angle.

Mirror planes are further classified as vertical (subscript 'v') or horizontal (subscript 'h') according to whether they contain the principal rotational axis of the molecule or are perpendicular to it.

As shown in the Figure, the C-H bond of chloroform is colinear with a C3 axis: each rotation of 360/3 = 120 degrees leaves the molecule in an identical arrangement.

The H-H molecule has a Cn of order infinity congruent with the H-H bond, and

Molecules may contain two other symmetry elements, which rarely need be specified.

The point group or symmetry group is the name given to the collection of symmetry elements possessed by a molecule. Each common collection of elements is represented by a simple symbol, called the Schonflies notation, indicating the type of reflection symmetry and the order of the principal rotational axis.

The following decision tree, based on a chart on p. 56 of Cotton, facilitates assignment of molecules to their point groups.

To use this chart, one simply answers "yes" or "no" to the questions posed in the boxes and follows the appropriate arrows to the point group designation. Let's make use of some examples.

For example (a), the answer to the first question, "Cn only?" is yes; the molecule has no reflection symmetry. The answer to the question, "Is n > 1?" is no. The molecule has only the trivial C1 axis possessed by all molecules, and hence belongs to point group C1.

Asking the first question about example (b) leads to a "no" answer. The plane of the virtual paper is a symmetry plane.

Allene, example (c), yields "no" answers to the first two questions. The red dotted line marks a C2 and the plane of the virtual paper is a s.

Example (d), benzene, likewise yields "no" answers to the first two questions. The C6 axis marked in red is evident, and the plane of the molecule, at least, is a symmetry plane.

Example (e) illustrates a molecule having only reflection symmetry; answering "no" to the first question and "yes" to the second places it in point group Cs. (The mirror plane bisects the molecule perpendicular to the ring plane, and contains the Cl-C-H plane.)

The planar anti conformation of hydrogen peroxide in (f) has a C2 perpendicular to the virtual paper (red dot) and the plane of the paper is a s. One answers "no", "no", "no", and "yes" to arrive at C2h for the point group.

Here are a few relationships that will help you in your search for symmetry elements:

Please refer to either of the references cited at the beginning of this page for a more complete analysis of symmetry properties and their relationship to constructing a valid set of molecular orbitals. (Orbital, rather than molecular, symmetries are usually represented with a different set of symbols, devised by Mulliken.)

Note that the Cs symmetry of the cyclopropane derivative above is the symmetry of Blake's tiger:

"Tiger, tiger, burning bright
In the forests of the night,
What immortal hand or eye
Shaped thy fearful symmetry?"

..... William Blake

It also is the symmetry (approximately) possesed by most of us critters who go about on the surface of the earth. In his book, "Climbing Mount Improbable", Richard Dawkins speculates on how this particular symmetry came to evolve. He argues that differentiating front from back follows from separating ingestion and excretory functions, and gravity provides the force to make top different from bottom. But no comparable mechanisms exist to differentiate left from right; hence, bilateral symmetry. Indeed, he suggests, a mutation that caused such a distinction would be harmful, since it might well cause the afflicted critter to move in circles!

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