# 22.4.1: i. Exercises

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## Q1

Consider the molecules $$CCl_4, CHCl_3, \text{ and } CH_2Cl_2.$$
a. What kind of rotor are they (symmetric top, ect; do not bother with oblate, or near-prolate, etc.)
b. Will they show pure rotational spectra?
c. Assume that ammonia shows a pure rotational spectrum. If the rotational constrants are 9.44 and 6.20 $$cm^{-1}$$, use the energy expression:
$E = (A - B) K^2 + B J(J + 1),$
to calculate the energies ( in $$cm^{-1}$$) of the first three lines (i.e., those with lowest K, J quantum number for the adsorbing level) in the absorption spectrum (ignoring higher order terms in the energy expression).

## Q2

The molecule $$^{11}B ^{16}O$$ has a vibrational frequency $$\omega_e = 1885 cm^{-1}$$, a rotational constant $$B-e = 1,78 cm^{-1}$$, and a bond energy from the bottom of the potential well of $$D_e^0 = 8.28 eV$$.

Use integral atomic masses in the following:
a. In the approximation that the molecule can be represented as a Morse oscillaor, calculate the bond length, $$R_e$$ in angstroms, the centrifugal distortion constant, $$D_e \text{ in } cm^{-1}$$, the anharmonicity constant, $$\omega_eX_e \text{ in cm}^{-1}$$, the zero-point corrected bond energy, $$D_0^0$$ in eV, the vibrational rotation interaction constant, $$\alpha_e \text{ in cm}^{-1}$$, and the vibrational state specific rotation constants, $$B_0 \text{ and } B_1 \text{ in cm}^{-1}$$. Use the vibration-rotation energy expression for a Morse oscillator:
\begin{align} E &=& & \hbar\omega_e \left(v + \dfrac{1}{2}\right) - \hbar \omega_eX_e\left( v = \dfrac{1}{2}\right)^2 + B_vJ(J + 1) - D_eJ^2(J + 1)^2 \text{, where} \\ B_v &=& & B_e - \alpha_e\left(v + \dfrac{1}{2}\right), \alpha_e = \dfrac{-6B_e^2}{\hbar\omega_e} + \dfrac{6\sqrt{B_e^3\hbar\omega_eX_e}}{\hbar\omega_e}\text{, and } D_e = \dfrac{4B_e^3}{\hbar\omega_e^2} \end{align}
b. Will this molecule show a pure rotation spectrum? A vibration-rotation spectrum? Assume that it does, what are the energies $$(in cm^{-1})$$ of the first three lines in the P branch $$(\Delta v = +1, \Delta J = -1)$$ of the fundamental absorption?

## Q3

Consider trans-$$C_2H_2Cl_2$$. The vibrational normal modes of this molecule are shown below. What is the symmetry of the molecule? Label each of the modes with the appropriate irreducible representation. This page titled 22.4.1: i. Exercises is shared under a not declared license and was authored, remixed, and/or curated by Jack Simons.