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Chapter 10: Problem 24
Determine all of the symmetry species spanned by the normal modes of chlorofluoromethane.
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Identify the conditions for the existence and locations of heads in the \(P\) - and \(R\) -branches of a diatomic molecule.
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 \(\psi(x) \psi(y) \propto \mathrm{e}^{\mathrm{jp}}\)
\( \mathrm{An}\) infrared spectrum of gaseous DCl revealed lines at the following wavenumbers for the lowest four transitions from the \(v=0\) state: 2091,4128,6111,8043 Determine the spectroscopic constants \(\omega\) and \(\omega x_{e^{-}}\).
The \(J+1 \leftarrow J\) rotational transitions of \(^{16} \mathrm{O}^{12} \mathrm{C}^{32} \mathrm{S}\) and \(^{16} \mathrm{O}^{12} \mathrm{C}^{34} \mathrm{S}\) occur at the following frequencies \((v / \mathrm{GHz})\) $$\begin{array}{lllll} \hline J & 1 & 2 & 3 & 4 \\ \hline^{16} \mathrm{O}^{12} \mathrm{C}^{32} \mathrm{S} & 24.32592 & 36.48882 & 48.65164 & 60.81408 \\ ^{16} \mathrm{O}^{12} \mathrm{C}^{34} \mathrm{S} & 23.73223 & & 47.46240 & \\ \hline \end{array}$$ Find (a) the rotational constants, (b) the moments of inertia, and (c) the CS and CO bond lengths. Hint. Begin by finding expressions for the moment of inertia \(I\) through \(I=m_{\Lambda} R_{A}^{2}+m_{B} R_{B}^{2}+m_{C} R_{C}^{2},\) where \(R_{X}\) is the distance of atom \(\mathrm{X}\) from the centre of mass. The easiest procedure is to use the result established in Exercise \(10.6,\) which leads to \(I=\left(m_{\mathrm{A}} m_{\mathrm{C}} / m\right)\left(R_{\mathrm{AB}}+R_{\mathrm{BC}}\right)^{2}+\left(m_{\mathrm{B}} / m\right)\left(m_{\mathrm{A}} R_{\mathrm{AB}}^{2}+m_{\mathrm{C}} R_{\mathrm{BC}}^{2}\right)\) The lengths \(R_{\text {AB }}\) and \(R_{\mathrm{BC}}\) may be found only if two values of \(I\) are known. Assume the bond lengths are the same in isotopomeric molecules.
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?
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