Practice Problem Set 4

1. A spectroscopic transition is observed at . Convert this value to wavelength, frequency and energy units.
2. Explain how the equilibrium bond length in a heteronuclear diatomic molecule may be measured.
3. Which of the following two trial wavefunctions gives the better approximation to the ground state wavefunction of a particle in a box?
   1. 
   2. 
4. Apply the variational method to the harmonic oscillator problem with the trial wavefunction with a>0. Should the variational energy obtained be a big surprise? Why or why not?
5. Using the trial wavefunction , obtain an approximation to the ground state energy of the quartic oscillator. For a quartic oscillator, .
6. Write down the Born-Oppenheimer electronic Hamiltonian for LiH. Use summation notation and write down all summation indices explicitly.
7. Explain the Born-Oppenheimer method for determining the energy levels of a diatomic molecule.
8. Briefly explain the self-consistent field method. Why is this method useful?
9. Suppose that electrons had a spin of instead of the usual .
   1. How many valence electrons would lithium have in this alternate Universe?
   2. Name the next element in lithium's group in this alternate Universe.
10. Rank the following reactions in the order in which you think tunneling would be most important. Explain your reasoning.
    1. 
    2. 
    3. 
11. Give a counterexample to the following proposition: If F and G are compatible observables, and G and H are compatible, then F is compatible with H.
12. Briefly explain how Fourier Transform (FT) NMR spectroscopy works.
13. Explain how spin-lattice relaxation times are measured by FT NMR spectroscopy.
14. Are kinetic energy and potential energy compatible observables for the one-dimensional harmonic oscillator?
15. The fundamental vibrational transition of occurs at . Calculate k for this molecule assuming harmonic-oscillator behavior. The isotopic masses of and are and , respectively.
16. A photon of wavelength 250nm is absorbed by a molecule with an approximate mass of (800g/mol). By how much does the velocity of the molecule change in the initial direction of travel of the photon?
17. If we write the bonding MO of a general diatomic AB as a superposition of one atomic orbital from each atom, we get the following set of equations:

    

    where and are the LCAO coefficients, , and are integrals involving the self-consistent field orbital Hamiltonian, S is the overlap integral, and is the orbital energy. Find the orbital energy . If there is more than one solution, explain how you would choose the best one given the values of , , and S.

18. How many normal modes of vibration would have?