Chapter 10

A diatomic molecule is found to have the following vibrational and rotational spectroscopic constants (all \\[ \left.\operatorname{in} \mathrm{cm}^{-1}\right): \tilde{v}=1525.25, \tilde{v} x_{\mathrm{e}}=21.74, \bar{B}_{\mathrm{e}}=8.295, \tilde{\alpha}_{\mathrm{e}}=0.186 \\] \(\bar{D}=0.325 ;\) the rotational constant depends on vibrational level as \(\tilde{B}_{v}=\tilde{B}_{e}-\left(\nu+\frac{1}{2}\right) \tilde{\alpha}_{e^{*}}\) If the diatomic molecule is initially in the state \((\nu=0, J=1),\) compute the wavenumbers of the \(R\) -and \(P\) -branch lines associated with the fundamental vibrational transition.

A diatomic molecule for which \(\tilde{v}=4401.2 \mathrm{cm}^{-1}\) and \(\bar{B}=121.3 \mathrm{cm}^{-1}\) is initially in the state \((\nu=1, J=2)\) In a Raman experiment utilizing \(15873.0 \mathrm{cm}^{-1}\) incident radiation, determine the wavenumber of the scattered radiation for (a) the Q-branch Stokes line, (b) the O-branch Stokes line, (c) the Q-branch anti-Stokes line. How will the wavenumber computed in part (a) change if the effects of anharmonicity are included?

Determine all of the symmetry species spanned by the normal modes of chlorofluoromethane.

Consider a two-dimensional harmonic oscillator with displacements in the \(x\) - and \(y\) -directions, the force constants being the same for each direction (the two bending modes of \(\mathrm{CO}_{2}\) is an example ). Show that the state resulting from the excitation of the oscillator to its first excited state can be regarded as possessing one unit of angular momentum about the \(z\) -axis. Hint. Show that \(\psi(x) \psi(y) \propto \mathrm{e}^{\mathrm{jp}}\)

Identify the conditions for the existence and locations of heads in the \(P\) - and \(R\) -branches of a diatomic molecule.