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Chemistry LibreTexts

Solutions 13

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  • Q1

    By looking at the modes of water, we can visually inspect and see if the symmetry operations result in staying the same (designated by a 1) or a reflection (-1), and then assign a representation that matches these results.

    Water Vibrational Mode \(E\) \(C_2\) \(\sigma_v(xz)\) \(\sigma_v(yz)\) Representation
    Symmetric Stretch 1 1 1 1 \(A_1\)
    Antisymmetric Stretch 1 -1 -1 1 \(B_2\)
    Scissoring Bend 1 1 1 1 \(A_1\)


    We use symmetry to evaluate integrals by comparing how symmetry operations affect 

    \[ M_{v \rightarrow v'} = \langle v' = 1 | \mu | v =0 \rangle\]

    The left side is unaffected but each factor of the right side is either changed by a factor of 1 or -1, which is determined by the character of the symmetry operation on the representation. If we represent a symmetry operation by \(\hat{R}\) , remember that the ground state has the representation \(A_1\), we have:

    \[\hat{R} M_{v \rightarrow v'} =   M_{v \rightarrow v'} = \hat{R}(\langle v' = 1 |) \hat{R}( \mu) \hat{R} (| v =0 \rangle) = \chi_{A_1}(\hat{R}) \chi_{\mu}(\hat{R}) \chi_{v=1}(\hat{R}) M_{v \rightarrow v'} \]

    We require

    \[\chi_{\mu}(\hat{R}) \chi_{v=1}(\hat{R}) =1 \]

    which occurs only if the representations of \(\mu\) and the excited state are the same. We can make a table to check how many combinations of \(\mu\) components and vibrational modes have the same representation, and therefore, an allowed transition:

    Mode\ Dipole Moment Component \(\mu_x\) \((B_1)\) \(\mu_y\)\((B_2)\) \(\mu_z\)\((A_1)\)
    Symmetric Stretch \((A_1)\)     same
    Antisymmetric Stretch \((B_2)\)   same  
    Scissoring Bend \((A_1)\)     same

    Only three combinations result in allowed transitions which include the x and y dipole moment components with different modes.