BCMB/CHEM 8190 PROBLEM SET 3 1) Write down a Hamiltonian for a pair of spin 1/2 spins in a static B0 field using the z component spin operators Iz1, Iz2, their coupling constant J12, and their offsets due to chemical shieldings, ν1 and ν2. Assume J is positive and ν1 < ν2. Make an energy level diagram showing the positions of levels for the four simple product spin functions. Show the allowed one quantum transitions and calculate the energy differences associated with them. Sketch and label the corresponding lines in a frequency resolved spectrum. 2) The 1H spectrum of the methyl-ethyl-ester of fumaric acid, CH3O(C=O)CH=CH(C=O)OCH2CH3, is highly second order showing olefinic resonances at 5.32, 5.35, 5.40, and 5.43 ppm at an operational frequency of 500 MHz. What is the H-H coupling constant between olefinic protons? At a qualitative level what would you expect the relative intensities of the four lines to be?