# 22.4.1: i. Exercises

- Page ID
- 84720

## 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.