Many molecules have significant symmetry in their structure. The use of symmetry can assist in simplifying calculations and in understanding concepts related to their properties. In order to understand how to systematically employ symmetry to classify and simplify calculations of molecular electronic structure and properties, we first consider the definitions of elements of symmetry.
Symmetry groups
We can orgznize a complete set of symmetry operations that describes a molecule into a group. A group is a self-consistent set of operations such that any product of two operations yields another operation in ths We can orgznize a complete set of symmetry operations that describes a molecule into a group. A group is a self-consistent set of operations such that any product of two operations yields another operation in ths group. We call the designation for a particular molecule its point group.
Construction of a character table
A character table consists of a the set of orthogonal basis vectors that define the space. Just as a Cartesian coordinate system is defined by orthogonal basis vectors along x, y and z, we can think of the point group as having a set of orthogonal basis vectors. One difference is that the dimension of the space of a point group is usually larger than 3. There as many basis vectors as symmetry operations i.e. both are equal to the dimension of the group. Still the basis vectors must be normalized and orthogoal. Thus, they consistitute a unique set. A second aspect of a a character table is the list of how certain Cartesian basis functions transform in the space of the point group. Linear functions, Rotations and Quadratic functions are sets of functions that are listed in columns to the right of the symmetry operations of the group. This is done for the convenience of the user. In theory one could determine these transformations based on a knowledge of the group. But, it is expedient to use the tabulated values in order to quickly establish which of the basis vectors (irreducible representations) in the group correspond to linear or quadratic Cartesian functions.
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