12.11: Reducible Representations are Comprised of Irreducible Representations
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As we saw in the previous section, the complete motion (translations, rotations, and vibrations) of ammonia (NH3) can be represented by the following reducible representation:
\(C_{3v}\) | \(E\) | \(2C_3\) | \(3\sigma_v\) |
---|---|---|---|
\(\Gamma\) | \(12\) | \(0\) | \(2\) |
We want to relate the reducible form representation to the irreducible representation. To do this, we use the tabular method. First, create a new table:
\(C_{3v}\) | \(E\) | \(2C_3\) | \(3\sigma_v\) |
---|---|---|---|
\(\Gamma\) | \(12\) | \(0\) | \(2\) |
\(A_1\) | |||
\(A_2\) | |||
\(E\) |
We will need the \(C_{3v}\) character table for ammonia:
\(C_{3v}\) | \(C_{3}\) | \(C_{s}\) |
---|---|---|
\(A_1\) | \(A\) | \(A'\) |
\(A_2\) | \(A\) | \(A"\) |
\(E\) | \(E\) | \(A'+A"\) |
Fill in each number in our table by using the following equation:
\[g_c \chi_i X_r\]
\(g_e\) | Number of operations (order) in the class |
\(\chi_i\) | Character of the irreducible representation from the character table |
\(\chi_r\) | Character of the reducible representation from \(\Gamma\) |
For example, the top-left value would be:
\[1\times 1\times 12=12\]
Where:
- \(1\) is the number of operations in the \(E\) class
- \(1\) is the character of the irreducible representation
- 9 is the character of the reducible representation
The table becomes:
\(C_{3v}\) | \(E\) | \(2C_3\) | \(3\sigma_v\) |
---|---|---|---|
\(\Gamma\) | \(12\) | \(0\) | \(2\) |
\(A_1\) | \(12\) | \(0\) | \(6\) |
\(A_2\) | \(12\) | \(0\) | \(-6\) |
\(E\) | \(24\) | \(0\) | \(0\) |
Sum up each row:
\(C_{3v}\) | \(E\) | \(2C_3\) | \(3\sigma_v\) | \(\sum\) |
---|---|---|---|---|
\(\Gamma\) | \(12\) | \(0\) | \(2\) | |
\(A_1\) | \(12\) | \(0\) | \(6\) | \(18\) |
\(A_2\) | \(12\) | \(0\) | \(-6\) | \(6\) |
\(E\) | \(24\) | \(0\) | \(0\) | \(24\) |
Now divide the summed values by the order of the group to obtain the number of times the irreducible representation appears (\(n_i\)). Ammonia has order \(h=6\):
\(C_{3v}\) | \(E\) | \(2C_3\) | \(3\sigma_v\) | \(\sum\) | \(n_i=\frac{\sum}{h}\) |
---|---|---|---|---|---|
\(\Gamma\) | \(12\) | \(0\) | \(2\) | ||
\(A_1\) | \(12\) | \(0\) | \(6\) | \(18\) | \(3\) |
\(A_2\) | \(12\) | \(0\) | \(-6\) | \(6\) | \(1\) |
\(E\) | \(24\) | \(0\) | \(0\) | \(24\) | \(4\) |
The reducible representation can be broken down to its irreducible forms:
\[\Gamma=3 A_1+A_2+4 E\]
Now that we have the irreducible representations for the motion of ammonia, we can determine which are associated with rotations, vibrations, and translations. To start, we turn to the \(C_{3v}\) character table:
\(C_{3v}\) | E | 2C3 | 3σv | ||
---|---|---|---|---|---|
A1 | 1 | 1 | 1 | z | x2+y2, z2 |
A2 | 1 | 1 | -1 | Rz | |
E | 2 | -1 | 0 | (Rx, Ry), (x,y) | (xz, yz) (x2-y2, xy) |
The first column to the right of the characters includes to the terms \(x\), \(y\), \(z\), \(R_x\), \(R_y\), and \(R_z\). The rows with \(x\), \(y\), and \(z\) represent the irreducible representations for the translational modes in those directions:
\[ \Gamma_\text{trans} = A_1 + E \]
The rows with \(R_x\), \(R_y\), and \(R_z\) represent the irreducible representations for the rotational modes about those axes:
\[ \Gamma_\text{rot} = A_2 + E \]
We can subtract the translational and rotational irreducible representations from our \Gamma to get the irreducible representations for the normal vibrational modes:
\[ \Gamma_\text{vib} = \Gamma - \Gamma_\text{trans} -\Gamma_\text{rot} \]
Doing this, we obtains:
\[ \Gamma_\text{vib} = 2 A_1 + 2 E \]