1. The bond potential of a diatomic molecule is given by,
\[ \begin{equation} U(R) = -\frac{A}{R^2}+\frac{B}{R^{10}} \end{equation} \]The bond length is 2.8 Å and the binding energy is 8 eV. Calculate the constants $A$ and $B$.
2. The energies associated with three normal modes of a molecule are 0.18678 eV, 1.05344 × 10-19 J, and 0.3752 eV. Calculate the frequencies and wavelengths of photons that are absorbed by these normal modes.
3. Consider the rotational states of a HF molecule. Calculate the rotational constant $B_e$ (Hint: $\hbar^2/2I=hcB_e$). What is the frequency of the photon that is absorbed when a HF molecule makes a transition from rotational state $J=2$ to rotational state $J=3$. What is the energy on the rotational state $J=4$? The bond length in a HF molecule is 92 pm and the masses of the atoms are $m_H=1$ u and $m_F=19$ u where u is the atomic mass unit: $\text{u}=1.66\times 10^{-27}$ kg.