A New Real-Space Finite Element Method to Solve the Kohn-Sham Equations of Density Functional Theory

基于Kohn-Sham密度泛函理论(DFT)的第一性原理量子力学模拟是现代材料研究的重要组成部分。该理论的无参数、量子力学的本质促进了对各种材料系统的基本理解和稳健的预测，从金属元素到绝缘有机。然而，求解DFT(耦合的薛定谔和泊松方程)方程是一项艰巨的任务，这严重限制了可以用这种严格的量子力学方法研究的材料系统的范围。当前最先进的DFT计算方法通过添加更多的网格点(有限差分法)或基函数(平面波和有限元方法)而扩展到更复杂的问题，而不考虑复杂性的性质，导致在处理高度不均匀的系统时效率极低，例如涉及第一行原子、过渡金属原子或鳗系元素的系统。该项目将克服当前方法的这一基本局限性，利用有限元分析中的单位划分技术将已知的原子物理构建到求解过程中，从而大大减少所需的自由度并增加可解决问题的规模。通过有效而准确地求解密度泛函理论方程，获得材料的电子结构和基本性质(机械、电、磁和光学)。由于其普遍性，所提出的电子结构计算的单位划分有限元方法有可能改变最大、最复杂的量子力学计算的方式，从而为金属、生物和纳米结构材料的新应用铺平道路，这是以前不可能的。与LLNL的John PASK博士的外部合作将导致开发一种优化的、完全自洽的实现，非常适合大规模并行计算平台。这种合作极大地加强了最大影响的可能性，实现了复杂模拟的更快计算时间，例如d电子和f电子系统的熔化、状态方程研究以及新材料中缺陷的结构和能量学。

First principles (ab initio) quantum mechanical simulations based on Kohn-Sham density functional theory (DFT) are a vital component of modern materials research. The parameter free, quantum mechanical nature of the theory facilitates both fundamental understanding and robust predictions across the gamut of materials systems, from metallic actinides to insulating organics.However, the solution of the equations of DFT (coupled Schrodinger and Poissonequations) is a formidable task and this has severely limited the range of materials systems that can be investigated by such rigorous, quantum mechanical means. Current state-of-the-art approaches for DFT calculations extend to more complex problems by adding more grid points (finite-difference methods) or basis functions (planewave and finite-element methods) without regard to the nature of the complexity, leading to substantial inefficiencies in the treatment of highly inhomogeneous systems such as those involving first-row, transition-metal or actinide atoms. This project will overcome this basic limitation of current approaches by employing partition-of-unity techniques in finite-element analysis to build the known atomic physics into the solution process, thus substantially reducing the degrees of freedom required and increasing the size of problems that can be addressed.The electronic structure and fundamental properties (mechanical, electrical, magnetic, and optical) of materials are obtained via efficient and accurate solutions of the equations of density functional theory. By virtue of its generality, the proposed partition-of-unity finite element method for electronic-structure calculations has the potential to change the way the largest, most complex quantum mechanical calculations are done, thereby paving the way for new applications in metallic, biological, and nanostructural materials that were not possible before. The external collaboration with Dr. John Pask at LLNL will lead to the development of an optimized, fully self-consistent implementation, well-suited to large-scale parallel computational platforms. This collaboration greatly strengthens the likelihood of maximum impact with the realization of faster computational times on complex simulations, such as melting of d- and f-electron systems, equation of state studies, and structure and energetics of defects in new materials.