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12.11: Reducible Representations are Comprised of Irreducible Representations

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    433848
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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 v    
    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:

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


    12.11: Reducible Representations are Comprised of Irreducible Representations is shared under a not declared license and was authored, remixed, and/or curated by Jerry LaRue.