12.11: Reducible Representations are Comprised of Irreducible Representations

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:

\[g_c \chi_i X_r \nonumber \]

\(g_e\)	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:

\[1\times 1\times 12=12 \nonumber \]

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:

\[\Gamma=3 A_1+A_2+4 E \nonumber \]

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:

\[ \Gamma_\text{trans} = A_1 + E \nonumber \]

The rows with \(R_x\), \(R_y\), and \(R_z\) represent the irreducible representations for the rotational modes about those axes:

\[ \Gamma_\text{rot} = A_2 + E \nonumber \]

We can subtract the translational and rotational irreducible representations from our \Gamma to get the irreducible representations for the normal vibrational modes:

\[ \Gamma_\text{vib} = \Gamma - \Gamma_\text{trans} -\Gamma_\text{rot} \nonumber \]

Doing this, we obtains:

\[ \Gamma_\text{vib} = 2 A_1 + 2 E \nonumber \]

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