Molecular Symmetry and Group Theory |
If a molecule has a proper rotation axis of order 3 or more, the application of some symmetry operations inevitably causes certain directional properties to be interconverted.
For example, the application of the C_{4} operation on the square planar molecule XeF_{4} [Point group?] converts a Xe p_{x} orbital into p_{y}, and p_{y} into -p_{x}. For the C_{4} operation to be valid the p_{x} and p_{y} orbitals must be identical and have the same energy i.e. they must be degenerate.
As the effect of the C_{4} operation on the p_{x} and p_{y} orbitals is not, for example, a simple inversion, it is impossible to describe by a single number. In this situation the p_{x} and p_{y} orbitals cannot be treated separately, but must be considered as a pair. Together these orbitals belong to a degenerate representation. |
If two directional properties are mixed by a symmetry operation, the effect of the operation can only be described by an array of numbers known as a matrix.
Following the rules of matrix multiplication (see inset), the effect of the matrix operating on the p_{x} and p_{y} orbitals is to convert p_{x} to p_{y} and p_{y} to -p_{x}: exactly the result of the C_{4} operation. Thus the matrix given describes the effect of the C_{4} operation. |
The sum of the elements on the leading diagonal (top left to bottom right i.e. a + d in the example above) of a square matrix is known as the character (or trace) of the matrix. It is this number that appears in character tables.
Note that the character is all that we need to apply group theory: we do not need the actual matrix.
The character of the matrix tells us to the extent to which the objects operated on (in this case the p_{x} and p_{y} orbitals) are ‘converted into themselves’ by the operation. As p_{x} is transformed into p_{y}, and p_{y} into -p_{x}, in this case the character is zero.
By convention doubly degenerate representations are labelled E and triply degenerate representations are labelled T.
In the point group D_{4h}, the p_{x} and p_{y} orbitals transform according to the representation labelled E_{u}. The subscript ‘u’ (for ungerade) indicates that it is anti-symmetric with respect to the inversion operation i.
The subscript ‘g’ (gerade) is used for representations that are symmetric to inversion.
The full set of symmetry operations for D_{4h}, and the characters of the E_{u} irreducible representation, are as follows:
D_{4h} | E | 2C_{4} | C_{2} | 2C_{2}^{′ } | 2C_{2}^{″} | i | 2S_{4} | σ_{h} | 2σ_{v} | 2σ_{d} | |
E_{u} | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | (x,y) |
Note that in character tables, degenerate directional properties (e.g. x and y) are bracketed together. This enables us to tell that certain orbitals such as p_{x} and p_{y} are degenerate. [Similarly in the D_{4h} point group, d_{xz} and d_{yz} are also degenerate.]
The characters in the table relate very simply to the extent to which the p_{x} and p_{y} orbitals are converted into themselves. Thus for C_{4} (p_{x} → p_{y}, p_{y} → -p_{x}) the character is zero, but for C_{2} about the same axis (p_{x} → -p_{x}, p_{y} → -p_{x}) the character is -2.
So far we have considered the effects of the C_{4} operation on only the Xe p_{x} and p_{y} orbitals. These orbitals are said to provide a basis for obtaining a representation of the C_{4} operation.
In general, a wide variety of different bases may be chosen depending on the nature of the problem being considered. For example, we may use the two O—H bonds in H_{2}O as a basis for the C_{2} operation of C_{2v}. As C_{2} converts O—H to O—H′ and vice versa, neither bond is converted into itself to any extent, and so the character of the matrix that describes this transformation must be zero. |
The σ(yz) operation, however, leaves both bonds unmoved, so its character must be 2.
[Similarly the character of the matrices describing E and σ(xz) are 2 and 0 respectively]