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Homework 13

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    In multiple times in the course so far, the three vibrations of water have been discussed (symmetric and asymmetric stretch and the bend). Identify the character (i.e., symmetric or antisymmetric) for how symmetry operations in the relevant point group of water operate on each vibration. Assign an irreducible representation for each vibration.


    The probability of an IR transition from the \(|v =0 \rangle \rightarrow |v' = 1 \rangle\) state is given by the transition moment integral

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

    where \(\mu\) is the electric dipole operator. The irreducible representations of \(\mu\) follows \(x\), \(y\) and \(z\) depending if the light is polarized in the \(x\)-, \(y\)-, \(z\)-directions, respectively. Given that the ground state-vibration wavefunction \(|v=0 \rangle\) is always the totally symmetry irreducible representation and excited-state vibration wavefunction \(|v'=1 \rangle\) is the symmetry of the vibration determined above in Q1.

    Use group theory and direct products to answer if each of the three vibrations of water will absorb like (i.e., not forbidden) with each of the three possible polarizations of light (Hint: you will have to use group theory to evaluate if nine transition moment integrals will be zero or not based on symmetry arguments of each of the three components in the integrand and direct product tables for the relevant point group).

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