The goal of this project is to investigate electronic structure models and their connections to macroscopic models from an applied mathematics perspective. The electronic structure models, in particular the density functional theory (DFT), are widely used to model materials and chemical systems from first principle. These models give rise to many interesting and challenging questions to applied mathematicians. The principal investigator focuses on the following aspects in this project: 1. Derivation of continuum theory of solid materials from electronic structure models. 2. Mathematical analysis of electronic structure models in the presence of the exchange-correlation energy. 3. Construction and analysis of force-based multiscale methods for electronic structure models. 4. Development of the frozen Gaussian approximation for electron dynamics in crystals and waves in periodic structures.
Electronic structure models are fundamental and indispensable tools in physics, chemistry, and material science. Further progress in this area needs better mathematical understanding of these models. This project develops more accurate models of materials from the quantum mechanical level. The understanding of mathematical properties of these models leads to more efficient algorithms to help extend the applicability of first principle calculations to macroscopic systems. Beyond the field of quantum chemistry, the proposed research contributes to related areas of applied mathematics, for example, applied nonlinear analysis, multiscale modeling and simulations, and numerical analysis. The investigator plans to combine research with education activities, including curriculum development to prepare students for interdisciplinary research.
