电子结构计算的数值方法与理论

第一原理电子结构计算已成为探索与研究物质机理、理解与预测材料性质的重要手段和工具.虽然第一原理电子结构计算取得了巨大的成功，但是如何利用高性能计算机又快又好地计算大规模体系，如何从数学角度理解电子结构模型的合理性与计算的可靠性和有效性，依然充满各种挑战.基于密度泛函理论的第一原理电子结构计算的核心数学模型为Kohn-Sham方程或相应的Kohn-Sham能量泛函极小问题.近年来，人们分别从非线性算子特征值问题的高效离散及Kohn-Sham能量泛函极小问题的最优化方法设计两个方面对电子结构计算的高效算法设计及分析展开了诸多研究.本文重点介绍我们小组在电子结构计算的方法与理论方面的一些进展，同时简单介绍该领域存在的困难与挑战.

NUMERICAL METHODS AND THEORIES FOR ELECTRONIC STRUCTURE CALCULATIONS

The first principles electronic structure calculations have become important tools for studying the material mechanism, understanding and predicting the material properties, and have achieved great success. However, it is still full of challenge for how to design highly efficient and highly accurate computational methods to deal with larger system, how to understand the reliability and efficiency of calculation from a mathematical point of view. Based on the Kohn-Sham DFT, the key mathematical modes for electronic structure calculations are the Kohn-Sham equation or the Kohn-Sham energy functional minimization problem. In the past decades, the highly efficient algorithms design and numerical analysis have attracted the attention of many distinguished mathematicians. Our group have also focused on this field and have done several works. In this paper, we introduce recent progresses in this field, mainly about those done by our group.