Mathematical Problems for Electronic Structure Models

这个项目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。从第一性原理出发，电子结构模型，特别是密度泛函理论(DFT)被广泛用于模拟材料和化学体系。这些模型给应用数学家带来了许多有趣和具有挑战性的问题。本课题的主要研究内容包括以下几个方面：1.从电子结构模型推导固体材料连续介质理论。对存在交换相关能的电子结构模型进行了数学分析。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、化学和材料科学中不可缺少的基本工具。这一领域的进一步进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了更精确的材料模型。对这些模型的数学性质的理解导致了更有效的算法，以帮助将第一性原理计算的适用性扩展到宏观系统。除了量子化学领域，拟议的研究还有助于应用数学的相关领域，例如应用非线性分析、多尺度建模和模拟以及数值分析。研究人员计划将研究与教育活动相结合，包括课程开发，为学生进行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.