论文标题
从统一的角度来看，电子结构方法中的结构化特征值问题
Structured eigenvalue problems in electronic structure methods from a unified perspective
论文作者
论文摘要
在（相对论）电子结构方法中，激发能量的四个矩阵特征值问题和线性响应（Bethe-Salpeter）特征值问题是两个经常遇到的结构化特征值问题。虽然对前者的问题进行了彻底的研究，但后来的问题以其最一般的形式，即，复杂的情况而不假设电子黑森的积极确定性，尚未完全理解。鉴于它们非常相似的数学结构，我们从统一的角度研究了这两个问题。我们表明，其特征向量的谎言组结构的识别为设计对角线化算法以及相应歧管上的数值优化技术提供了一个框架。通过为四个矩阵特征值问题使用相同的还原算法，我们提供了一种必要和足够的条件来表征不同的情况，其中原始线性响应特征值问题的特征值是真实的，纯粹是虚构的，或复杂的。结果可以看作是对真实矩阵案例众所周知条件的自然概括。
In (relativistic) electronic structure methods, the quaternion matrix eigenvalue problem and the linear response (Bethe-Salpeter) eigenvalue problem for excitation energies are two frequently encountered structured eigenvalue problems. While the former problem was thoroughly studied, the later problem in its most general form, namely, the complex case without assuming the positive definiteness of the electronic Hessian, is not fully understood. In view of their very similar mathematical structures, we examined these two problems from a unified point of view. We showed that the identification of Lie group structures for their eigenvectors provides a framework to design diagonalization algorithms as well as numerical optimizations techniques on the corresponding manifolds. By using the same reduction algorithm for the quaternion matrix eigenvalue problem, we provided a necessary and sufficient condition to characterize the different scenarios, where the eigenvalues of the original linear response eigenvalue problem are real, purely imaginary, or complex. The result can be viewed as a natural generalization of the well-known condition for the real matrix case.
