In symmetry analysis, a representation is a way of describing how a set of objects (coordinates, vectors, orbitals, or motions) responds to the symmetry operations of a point group. Representations provide the bridge between symmetry elements and the physical behavior of molecules.
1. Group multiplication tables
A group multiplication table shows the result of performing two symmetry operations in sequence. The entry at the intersection of row \(A\) and column \(B\) is the operation obtained by applying \(B\) first, then \(A\).
For the point group \(C_{2v}\), the symmetry operations are \(E, C_2, \sigma_v(xz), \sigma_v'(yz)\). Every operation combined with any other operation yields another operation in the same set, demonstrating that the group is closed under multiplication. A similar idea applies to \(C_{3v}\), which contains the operations \(E, 2C_3, 3\sigma_v\). Below are the group multiplcation tables for the groups C2v and C3v.
\[ \begin{array}{c|cccc} & E & C_2 & \sigma_v & \sigma_v' \\ \hline E & E & C_2 & \sigma_v & \sigma_v' \\ C_2 & C_2 & E & \sigma_v' & \sigma_v \\ \sigma_v & \sigma_v & \sigma_v' & E & C_2 \\ \sigma_v' & \sigma_v' & \sigma_v & C_2 & E \end{array} \]
\[ \begin{array}{c|cccccc} & E & C_3 & C_3^2 & \sigma_v & \sigma_v' & \sigma_v'' \\ \hline E & E & C_3 & C_3^2 & \sigma_v & \sigma_v' & \sigma_v'' \\ C_3 & C_3 & C_3^2 & E & \sigma_v' & \sigma_v'' & \sigma_v \\ C_3^2 & C_3^2 & E & C_3 & \sigma_v'' & \sigma_v & \sigma_v' \\ \sigma_v & \sigma_v & \sigma_v'' & \sigma_v' & E & C_3^2 & C_3 \\ \sigma_v' & \sigma_v' & \sigma_v & \sigma_v'' & C_3 & E & C_3^2 \\ \sigma_v'' & \sigma_v'' & \sigma_v' & \sigma_v & C_3^2 & C_3 & E \end{array} \]
Group multiplication tables confirm that symmetry operations form a mathematical group.
2. A simple representation for \(C_{2v}\)
Consider the Cartesian coordinates \(x, y, z\) in a molecule belonging to \(C_{2v}\). Each symmetry operation either leaves a coordinate unchanged or changes its sign. We can represent this behavior using numbers.
For example, the coordinate \(z\) transforms as:
- \(E:\; z \rightarrow z\) (unchanged)
- \(C_2:\; z \rightarrow z\)
- \(\sigma_v:\; z \rightarrow z\)
- \(\sigma_v':\; z \rightarrow z\)
Similarly for the x and y axes:
- \(E:\; x \rightarrow x\) (unchanged)
- \(C_2:\; x \rightarrow -x\)
- \(\sigma_v:\; x \rightarrow x\)
- \(\sigma_v':\; x \rightarrow -x\)
- \(E:\; y \rightarrow y\) (unchanged)
- \(C_2:\; y \rightarrow -y\)
- \(\sigma_v:\; y \rightarrow -y\)
- \(\sigma_v':\; y \rightarrow y\)
Assigning \( +1 \) for unchanged behavior and \( -1 \) for sign reversal gives one-dimensional representations, each derived for how a given axis tranforms under the group operations:
\[ \Gamma(z) = (1,\;1,\;1,\;1) \] \[ \Gamma(x) = (1,\;-1,\;1,\;-1) \] \[ \Gamma(y) = (1,\;-1,\;-1,\;1) \]
These representations correspond to three (of the four) irreducible representations of \(C_{2v}\).
3. A reducible representation for \(C_{2v}\)
Now consider the set of Cartesian coordinates \( (x, y, z) \) together. Each symmetry operation acts on all three coordinates, and the combined effect can be summarized by the character, which is the trace of the transformation.
Counting how many coordinates remain unchanged under each operation gives:
\[ \Gamma_{\text{cart}} = (3,\; -1,\; 1,\; 1) \]
This is a reducible representation, because it can be written as a sum of simpler (irreducible) representations. Reducible representations arise naturally from sets of vectors, atomic motions, and basis functions.
4. An irreducible representation for \(C_{3v}\)
In \(C_{3v}\), not all irreducible representations are one-dimensional. A particularly important example is the two-dimensional irreducible representation \(E\).
The pair of coordinates \( (x, y) \) transforms together under the symmetry operations of \(C_{3v}\). Rotations by \(120^\circ\) mix \(x\) and \(y\), rather than leaving them independent.
The characters for this irreducible representation are:
\[ \Gamma_E = (2,\; -1,\; 0) \]
where the entries correspond to the classes \(E, 2C_3, 3\sigma_v\). In this case, the representation of each symmetry operation in C3v requires a 2x2 matrix. (There are also two representations that are possible using just 1 and -1.) However, because the x and y axes are not eigenfunctions of the symmetry operations of the C3v point group, they must be used as a pair to form a basis for one of the irreducible representation. This representation cannot be decomposed further and is therefore irreducible.
Irreducible representations of \(C_{3v}\)
The point group \(C_{3v}\) has three irreducible representations: two 1D representations (\(\Gamma_1\) and \(\Gamma_2\)) and one 2D representation (\(\Gamma_3\)).
\(\Gamma_1\) (1D)
\[ \Gamma_1(E)=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_1(C_3)=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_1(C_3^2)=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_1(\sigma_v)=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_1(\sigma_v')=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_1(\sigma_v'')=\begin{pmatrix}1\end{pmatrix} \]
\(\Gamma_2\) (1D)
Rotations leave the sign unchanged, while reflections introduce a sign change.
\[ \Gamma_2(E)=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_2(C_3)=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_2(C_3^2)=\begin{pmatrix}1\end{pmatrix},\quad \Gamma_2(\sigma_v)=\begin{pmatrix}-1\end{pmatrix},\quad \Gamma_2(\sigma_v')=\begin{pmatrix}-1\end{pmatrix},\quad \Gamma_2(\sigma_v'')=\begin{pmatrix}-1\end{pmatrix} \]
\(\Gamma_3\) (2D)
One convenient choice of basis is the pair \((x,y)\). In this basis, rotations mix \(x\) and \(y\), so the matrices are \(2\times2\).
\[ \Gamma_3(E)= \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \]
\[ \Gamma_3(C_3)= \begin{pmatrix} -\tfrac{1}{2} & -\tfrac{\sqrt{3}}{2}\\ \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} \end{pmatrix}, \qquad \Gamma_3(C_3^2)= \begin{pmatrix} -\tfrac{1}{2} & \tfrac{\sqrt{3}}{2}\\ -\tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} \end{pmatrix} \]
\[ \Gamma_3(\sigma_v)= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, \qquad \Gamma_3(\sigma_v')= \begin{pmatrix} -\tfrac{1}{2} & \tfrac{\sqrt{3}}{2}\\ \tfrac{\sqrt{3}}{2} & \tfrac{1}{2} \end{pmatrix}, \qquad \Gamma_3(\sigma_v'')= \begin{pmatrix} -\tfrac{1}{2} & -\tfrac{\sqrt{3}}{2}\\ -\tfrac{\sqrt{3}}{2} & \tfrac{1}{2} \end{pmatrix} \]
Note that characters shown for Γ(E) above are generated by adding the diagonal elements of the matrices representing Γ(3). This generates the same value for the two C3 operations, as well as the same value for the three σv operations. This property simplifies things, as the characters of symmetry elements that belong to the same class will always have the same character! (Now how much would you pay?)
Also Note: any set of matrices related by a change of basis represents the same irrep. What matters physically are the symmetry properties encoded by the representation.
Big idea: representations encode how objects transform under symmetry operations. Reducible representations describe composite behavior, while irreducible representations are the fundamental building blocks used in spectroscopy and molecular orbital theory.