4: Molecular Symmetry and Point Groups

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Symmetry in Chemistry

Symmetry is actually a concept of mathematics and not of chemistry. However, symmetry, and the underlying mathematical theory for symmetry, group theory, are of tremendous importance in chemistry because they can be applied to many chemistry problems. For example it helps us to classify the structures of molecules and crystals, understand chemical bonding, predict vibrational spectra, and determine the optical activity of compounds. We will therefore first discuss the general foundations of symmetry and group theory, and then how they can be applied to chemical problems.

Let us first find a definition for symmetry. The human brain innately recognizes symmetry and patterns and we associate symmetry with beauty, but very familiar things are not necessarily easy to define scientifically. One common definition is that symmetry is the self-similarity of an object. The more similar parts it has the more symmetric it appears. For example, we would argue that the two wings of the butterfly depicted look similar. If the left wing was very different from the right wing the butterfly would look less symmetric.

Figure \(\PageIndex{1}\): Symmetry depicted through an image of a butterfly (Photo by Alfred Schrock on Unsplash)

In other classes you have learned different ways to describe symmetry, radial and bilateral symmetry in biology; R and S enantiomers or cis and trans isomers in organic chemistry. Group theory gives us a universal language and tools to define and describe the symmetry of any object or molecule.

This TED talk explains symmetry from a mathematical and artistic perspective.

Learning Objectives

Identify symmetry elements and operations in molecules and objects
Use flowchart to assign a molecule to a point group
Recognize the components of a character table
Determine whether a vibrational mode is infrared allowed

Thumbnail image shows all the reflection planes in ferrocene. Image from Otterbein University symmetry gallery

4.1: Symmetry Operations and Elements
The symmetry of a molecule consists of symmetry operations and symmetry elements. A symmetry operation is an operation that is performed to a molecule which leaves it indistinguishable and superimposable on the original position. Symmetry operations are performed with respect to symmetry elements (points, lines, or planes).
4.2: Point Groups
In group theory, molecules or other objects can be organized into point groups based on the type and number of symmetry operations they possess. Every molecule in a point group will have all of the same symmetry operations as any other molecule in that same point group.
4.3: Introduction to Character Tables
Although the method of assigning a point group to a molecule depends on some knowledge of the symmetry elements the molecule has, it does not require the consideration of all elements. This is where the character table of the point group comes into play.
4.4: Applications of Symmetry in Chemistry
There are many ways that symmetry and group theory can be applied to chemical problems. Examples include determining orbital overlap in molecular orbital theory and determining spectroscopic selections rules.