Tracy Huggins, a double major in physics and mathematics, was awarded a summer SMART grant from IU South Bend to research matrix eigenvalues problems with Professor Marmorino, our physical chemist. Junior and senior chemistry majors might recognize the matrix shown as very similar to the ones used in Huckel Molecular Orbital (HMO) theory.
In HMO theory the gamma parameter is replaced by the number one and each "1" indicates the energetic interaction of a carbon 2p-orbital with a parallel 2p-orbital on an adjacent carbon. The eigenvalues of an N-dimensional matrix approximate the energy levels of the conjugated electrons of a molecule with N carbons involved in alternating single and double bonds - when gamma equals one, that is. When gamma is zero, there is no cooperation between the double bonds and each acts like that of an independent ethene molecule.
Huggins has been working with Marmorino to get explicit expressions for the eigenvalues of arbitrarily sized-matrices when gamma lies between zero and one - between the simple limits of no conjugation and complete conjugation. The plan is to use these eigenvalue expressions to relate the parameter gamma to the energy difference of an electronic transition - and thus wavelength of light. This wavelength can be measured spectroscopically and then gamma can be determined and insight into the amount of conjugation is gained. Marmorino hopes to incorporate the results of this research into a physical chemistry experiment for undergraduates to replace a traditional one in which the wavelength of the transition is used to estimate the length of the carbon chain by applying the particle-in-a-box quantum model.
This research has given Huggins and Marmorino many surprises. It was relatively easy to obtain exact expressions for matrices of odd dimension, but we have found that the even dimensional matrices do not reveal exact solutions. It is quite interesting that the difficulty lies not in the size of the matrix, but rather where its dimension is even or odd. In the search for ways to approximate the elusive eigenvalues, Huggins has unearthed many mathematical theorems and delved into complex analysis.