Mathematical Problems for Electronic Structure Models

本课题的目标是从应用数学的角度研究电子结构模型及其与宏观模型的联系。电子结构模型，特别是密度泛函理论（DFT），从第一性原理出发被广泛用于材料和化学系统的建模。这些模型给应用数学家提出了许多有趣和具有挑战性的问题。项目负责人主要关注以下几个方面：1。从电子结构模型推导固体材料的连续介质理论。交换相关能存在下电子结构模型的数学分析。基于力的电子结构模型多尺度方法的构建与分析。晶体中电子动力学和周期结构中的波的冻结高斯近似的发展。电子结构模型是物理、化学和材料科学中必不可少的基础工具。这一领域的进一步发展需要对这些模型有更好的数学理解。该项目从量子力学层面开发更精确的材料模型。对这些模型的数学性质的理解会导致更有效的算法，以帮助将第一性原理计算的适用性扩展到宏观系统。除量子化学领域外，本研究还对应用数学的相关领域，如应用非线性分析、多尺度建模与模拟、数值分析等做出了贡献。研究者计划将研究与教育活动结合起来，包括课程开发，以使学生为跨学科研究做好准备。

Lu1312659 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.