Lu Yangyi, Gao Jiali
Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, Shenzhen 518055, China.
Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States.
J Chem Theory Comput. 2024 Oct 8;20(19):8474-8481. doi: 10.1021/acs.jctc.4c00545. Epub 2024 Sep 24.
Beyond the Hohenberg-Kohn density functional theory for the ground state, it has been established that the Hamiltonian matrix for a finite number () of lowest eigenstates is a matrix density functional. Its fundamental variable─the matrix density ()─can be represented by, or mapped to, a set of auxiliary, multiconfigurational wave functions expressed as a linear combination of no more than determinant configurations. The latter defines a minimal active space (MAS), which naturally leads to the introduction of the correlation matrix functional, responsible for the electronic correlation effects outside the MAS. In this study, we report a set of rigorous conditions in the Hamiltonian matrix functional, derived by enforcing the symmetry of a Hilbert subspace, namely the subspace invariance property. We further establish a fundamental theorem on the correlation matrix functional. That is, given the correlation functional for a single state in the -dimensional subspace, all elements of the correlation matrix functional for the entire subspace are uniquely determined. These findings reveal the intricate structure of electronic correlation within the Hilbert subspace of lowest eigenstates and suggest a promising direction for efficient simulation of excited states.
除了针对基态的 Hohenberg-Kohn 密度泛函理论外，已经确定，对于有限数量（）的最低本征态，哈密顿矩阵是一个矩阵密度泛函。其基本变量——矩阵密度（）——可以由一组辅助的多组态波函数表示或映射到该组函数上，这些波函数被表示为不超过 行列式构型的线性组合。后者定义了一个最小活性空间（MAS），这自然导致了相关矩阵泛函的引入，该泛函负责 MAS 之外的电子相关效应。在本研究中，我们报告了通过强制希尔伯特子空间的对称性，即子空间不变性，在哈密顿矩阵泛函中得出的一组严格条件。我们进一步建立了关于相关矩阵泛函的一个基本定理。也就是说，给定 - 维子空间中单个态的相关泛函，整个子空间的相关矩阵泛函的所有元素都被唯一确定。这些发现揭示了最低本征态的希尔伯特子空间内电子相关的复杂结构，并为激发态的高效模拟提出了一个有前景的方向。
