7: Group Theory and Symmetry
- Page ID
- 467611
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- 7.1: Symmetry Elements and Operations Define the Point Groups
- A symmetry operation is an action that leaves an object looking the same after it has been carried out. Each symmetry operation has a corresponding symmetry element, which is the axis, plane, line or point with respect to which the symmetry operation is carried out. The symmetry element consists of all the points that stay in the same place when the symmetry operation is performed. Point groups are defined by the total the symmetry operations that belong to that group.
- 7.2: Symmetry Operations Define Groups
- A mathematical group is defined as a set of elements (\(g_1\), \(g_2\), \(g_3\)...) together with a rule for forming combinations \(g_j\). The number of elements \(h\) is called the order of the group. For our purposes, the elements are the symmetry operations of a molecule and the rule for combining them is the sequential application of symmetry operations investigated in the previous section.
- 7.3: Symmetry Operations as Matrices
- We can carry out the various symmetry operations in group theory using transformation matrices. Transformation matrices are matrices that map one set of coordinates or functions onto another set.
- 7.5: Character Tables Summarize the Properties of a Point Group
- Now that we have learned how to create a matrix representation of a point group within a given basis, we will move on to look at some of the properties that make these representations so powerful in the treatment of molecular symmetry.
- 7.6: Characters of Irreducible Representations
- the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form.
- 7.8: H₂O
- Water is a bent molecule, and so it is important to remember that interactions of pendant ligands are dependent on their positions in space. You should consider the positions of the three atoms in water to be essentially fixed in relation to each other. The process for constructing the molecular orbital diagram for a non-linear molecule, like water, is similar to the process for linear molecules. We will walk through the steps below to construct the molecular orbital diagram of water.
- 7.9: The Exploitation of Symmetry Can Help Simplify Numerical Calculations
- Molecular symmetry can be used to simplify numerical calculations and provide qualitative information about vibrational modes, electronic structure, and more. The process of obtaining this information is based on group theory.