## Daniel Jensen and Dr. Brett Hess, Physics and Astronomy

Quantum mechanics is a branch of physics that is used to explain the physical properties of small molecules, nanoclusters, and other atomic systems. The principal equation used in quantum mechanics is the Schrödinger equation and its solution for a particular system tells us everything that can be known about that system’s physical properties. A major goal of our research group is to approximately solve the Schrödinger equation using computational methods. In particular we solve for the excited-state properties of the system. This then allows us to analyze optical properties in nanoclusters such as photocatalysis in TiO2.

My main contribution to this project was programming some of the integrals that describe the interactions between two electrons orbiting different atoms. Specifically, I computed the four-orbital integrals containing the Coulomb kernel that are needed for most excited-state calculations. These integrals were computed for typical molecular bonding distances and stored for later use in a molecular dynamics program called FIREBALL. I then assisted a graduate student in our group to use these integrals in our new code for FIREBALL based on time-dependent density functional theory and Görling and Levy perturbation theory.

One of the greatest challenges I encountered while programming the four-orbital integrals was choosing how to perform the analytical and numerical integrations required for each integral. There are many ways to perform the analytical integrations and although my initial derivations were correct, the resulting integrals required more storage space and time to complete than those based on my final derivations. Also, the numerical integration routine based on Simpson’s Rule that I programmed initially was very slow and imprecise compared to the integration routine I later implemented based on the recursive Simpson’s Rule.

As a result of this research I developed several skills that are not usually needed to do textbook problems but are essential for doing research. For example, I learned how to scale units in equations, collaborate and share code in a team project, and present results to a group. Due to the nature of this project I learned a lot about other fields besides physics such as chemistry, numerical analysis, and computer science. For example, I read several papers as well as a book written by chemists to understand the basics of density functional theory. I also read a lot of numerical analysis books to learn about error estimates and numerical stability for improving our integration routines. Additionally, I studied a lot of programming manuals to learn about sharing code, writing modular programs, and creating parallel programs to be run on supercomputers.

I presented our initial findings at the Spring Research Conference in March 2007. I then wrote and submitted a senior thesis on our final results of the four-orbital integrals. This thesis will be stored by the BYU Physics Department. I have since started a master’s program in physics at BYU and am continuing our research by adding more molecular dynamics capabilities to our program. Now that most of our program is written and debugged, we are beginning to apply it to different nanoclusters such as TiO2 so that we can better understand their optical properties.