Mathematical Studies in Quantum Mechanics

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

This project concentrates primarily on the investigation of mathematical problems that arise in the study of molecular quantum mechanics. The nuclei in a molecule have large masses. Consequently, their motion is approximately described by classical mechanics. By comparison, electron masses are very small, and electrons adjust their quantum mechanical states rapidly in response to the much slower motion of the nuclei. This intuition is the basis for Born-Oppenheimer approximations. Professor Hagedorn, his graduate students, and his collaborators will study situations where Born-Oppenheimer approximations are not sufficiently accurate to describe all the phenomena of interest. For example, they plan to develop accurate "surface hopping models" to include the possibility of electrons making quantum mechanical transitions. Models of this type have been proposed in the chemistry and physics literature, but they have not been derived from first principles. The main goal of this project is rigorously to derive and implement such a model. Motion of electrons and nuclei in molecules are well described by solutions to the Schrodinger equation. No one has ever solved the Schrodinger equation for a molecule, and our theoretical knowledge of chemistry and molecular physics has come from approximations to solutions. The principal goals of this project are to study the several commonly used approximations and to develop improved approximations that will be useful to chemists and physicists. In addition to theoretical understanding, potential uses of this basic research are the synthesis of new chemical compounds, the design of new catalysts, or the design of new drugs.