12.11: Reducible Representations are Comprised of Irreducible Representations
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- 433848
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[ \Gamma_\text{vib} = 2 A_1 + 2 E \]