Fias Stijn, Ayers Paul W, De Proft Frank, Geerlings Paul
McMaster University, 1280 Main St. W, Hamilton, Ontario L8S 4L8, Canada.
Research Group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, B-1050 Brussels, Belgium.
J Chem Phys. 2022 Sep 21;157(11):114102. doi: 10.1063/5.0094653.
An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato's theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed.
给出了正则系综、巨正则系综、同构系综和巨同构系综的非局部二阶导数的数学性质概述。讨论了它们的半正定性或半负定性的意义以及这些性质对原子和分子的影响。基于此性质，可以推导出许多其他有趣的性质，例如本征函数展开、对角元和非对角元的界以及这些核的本征值。我们还证明了软度核和线性响应的加藤定理以及当将一个系统解离成两个非相互作用的分子片段时线性响应的解离极限等于各个片段线性响应之和。最后，讨论了这些核、它们的本征函数及其本征值的实际计算策略。
