CHEM 442 Lecture 23 Problems
23-1. Write down the Schrödinger equation for the hydrogen molecule. Specify the range
of summation indexes.
23-2. Write down the Schrödinger equation for the benzene molecule. Specify the range
of summation indexes.
23-3. Write down the electronic Schrödinger equation in the Born–Oppenheimer
approximation. Which coordinates are parameters and which are variables?
23-4. Write down the nuclear Schrödinger equation in the Born–Oppenheimer
approximation. What is the potential energy surface?
23-5. Show mathematically that, in the limit that all nuclear masses tend to infinity, the
molecular Hamiltonian reduces to an electronic Hamiltonian, which differs from the
electronic Hamiltonian of 23-3 merely by a constant.
23-6. Explain how the concepts of equilibrium structure, dissociation energy, molecular
vibrations, and molecular rotations derive from the Born–Oppenheimer approximation.
23-7. Draw the potential energy curves of the hydrogen molecule in the ground state
(stable) and in the first excited state (unstable towards dissociation) as a function the
atom-atom distance. Locate equilibrium bond lengths and draw vibrational energy levels.
23-8. Draw the potential energy curves of the water molecule in the ground state (stable)
as a function of HOH bond angle.
23-9. What are the electronic, translational, vibrational, and rotational degrees of freedom
in the hydrogen molecule?
23-10. What are the electronic, translational, vibrational, and rotational degrees of
freedom in the benzene molecule?