Efficient Algorithms for Electronic Structure Analysis

本提案的主要目标是开发适用于金属和绝缘体的有效的电子结构分析算法。这将通过发展费米算符的多极表示来实现，费米算符是电子结构分析和更广泛的物质量子理论中的一个基本对象。此外，PI还建议开发有效的算法来表示和计算在此背景下出现的格林函数。该项目的第二部分是对电子结构分析中的算法进行数值分析。将研究的课题包括线性定标算法的精度、自洽迭代的收敛和收敛速度。这将通过开发和使用简单但规范的模型问题来实现，这些问题捕捉到问题的基本方面，但允许显式的分析计算。电子结构分析是化学和材料科学以及生物学的某些方面的基础。我们理解化学反应和材料基本方面的能力在很大程度上依赖于从量子化学或密度泛函理论中求解模型的有效数值算法。现有的算法对绝缘体比对金属更有效。对于最近开发的线性定标算法尤其如此，该算法严重依赖于波函数或密度矩阵的指数衰减特性，这一特性适用于绝缘体，但不适用于金属。本项目的目的是将强大的数学工具应用于电子结构分析问题。这项拟议的工作将以一种前所未有的方式探索电子结构问题的数学特征。通过这样做，新的见解和新的算法将极大地提高我们分析物质电子结构的能力。

The main objective of the present proposal is to develop efficient algorithms for electronic structure analysis that are applicable for both metals and insulators. This will be done by developing multipole representation of the Fermi operator, which is a fundamental object in electronic structure analysis and more generally quantum theories of matter. In addition, the PI also proposes to develop efficient algorithms for representing and computing the Green's functions that arise in this context. A second component of the project is the numerical analysis of the algorithms in electronic structure analysis. Topics to be studied include the accuracy of linear scaling algorithms, convergence and convergence rates of self-consistent iterations. This will be done by developing and using simple but canonical model problems that capture the essential aspects of the problem but allow explicit analytical calculations.Electronic structure analysis is at the foundation of chemistry and material science, as well as some aspects of biology. Our ability to understand chemical reactions and fundamental aspect of materials relies heavily on efficient numerical algorithms for solving models from quantum chemistry or density functional theory. Existing algorithms are much more effective for insulators than for metals. This is particularly true for the recently developed linear scaling algorithms which relies heavily on the exponential decay property of the wave functions or density matrices, a property that holds for insulators but not for metals. The present project is aimed at bringing powerful mathematical tools to bear on the problem of electronic structure analysis. The proposed work will explore the mathematical features of the electronic structure problem in a way that has never been done before. By doing so, new insights and new algorithms will result that greatly advance our ability to analyze the electronic structure of matter.