Chemistry 3730
Fall 1995 Final Examination

Aggregate value of this examination: 40 marks.

Section 1

Answer all questions in this section. Aggregate value of questions in this section: 24.

1. In intense laser fields, two-photon processes sometimes occur. In a two-photon process, two photons are simultaneously absorbed by a molecule (hence the name).
   1. State an essential condition for the simultaneous absorption of two photons by a molecule.
   2. Derive the selection rule for the change in the electronic angular momentum quantum number during a two-photon process.
2. For a two-level system in a radiation field, the wavefunction can be written as a time-dependent superposition of the pure states and  :

   

   The coefficients and evolve according to

   

   In these equations, is the perturbation induced by the radiation field. Explain what these equations tell us about absorption and emission of radiation by matter.

3. 1. State the Bloch theorem.
   2. Explain why the Krönig-Penney model leads to bands of allowed electronic energies.
   3. Distinguish electrical insulators and conductors on the basis of the band theory.
4. Describe in detail a spectroscopic method for measuring a molecular parameter. You may choose any parameter in any type of molecule but you must specify the type of spectroscopic information required and relate this as precisely as possible (using equations if possible) to the parameter you have chosen. Make sure to accurately define the parameter whose measurement you will describe.
5. 1. The fluorescence lifetime of a certain chemical group is measured and found to be 8ns. A quencher is added at a concentration of 0.2mol/L and the fluorescence lifetime decreases to 3ns. What is the rate constant for the bimolecular quenching process? 
   2. A chemical reaction is carried out between the fluorescent molecule and quencher of question 5a whose product contains both the fluorescent (donor) and quenching (acceptor) groups. A solution of this chemically modified molecule is prepared. If the fluorescence lifetime of the donor is 1ns, what is the rate constant for intramolecular quenching? (In class, we called this process excitation transfer.)
   3. Set and calculate , the effective intramolecular quencher concentration. What information does this number convey?

Section 2

Answer one of the following two questions, for six marks.

1. 1. Give the Hamiltonian and boundary conditions for the particle-in-a-box problem.
   2. Apply the variational method to the particle-in-a-box problem using a sine function as your trial wavefunction. Do you get all of the particle-in-a-box solutions by this method? Discuss.
2. Using the trial wavefunction , compute the approximate ground state energy of the one-dimensional quartic oscillator with Hamiltonian

   

   The following integral will be useful:

   

Section 3

Answer one of the following two questions, for ten marks.

1. 1. Write down the Born-Oppenheimer electronic Hamiltonian for LiH. If you use summation symbols in your Hamiltonian (a practice which I recommend), make sure to write down the summation limits explicitly.
   2. Write down the ground state electronic wavefunction of LiH as a Slater determinant of molecular orbitals.
   3. Write down a Slater determinant for an excited state of LiH. How does it differ from the ground state Slater determinant?
   4. Explain how you would go about determining approximate formulas for the LiH molecular orbitals at a particular internuclear separation R.
   5. Explain why I specified ``at a particular internuclear separation'' in the previous question in relation to Born-Oppenheimer theory.
2. Describe in detail a method for obtaining an accurate ground-state energy for the helium atom. Show all relevant equations and discuss any difficulties which you anticipate in their solution. Also briefly describe how excited state energies might be obtained.

Useful fact

For a Hermitian operator ,