在预算上探索希尔伯特空间：强相关区域中测试电子结构方法的新基准集和性能度量
Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation
摘要：
本工作探讨了经典的电子结构方法的能力，有效地表示（压缩）的信息内容的全组态相互作用（FCI）波函数。我们介绍了一套基准的四个氢模型系统的不同维度和独特的电子结构：一维链，一维环，二维三角晶格，和三维密堆积金字塔。为了评估计算方法产生准确和紧凑的波函数的能力，我们引入了准确量，一个度量，衡量实现目标能量误差所需的变分参数的数量。使用这个度量和氢模型，我们研究了三种经典确定性方法的性能：i）通过最重要的决定子的后验和变分选择实现的选择构型相互作用（sCI），ii）通过FCI张量的后验奇异值分解（SVD-FCI）获得的秩降低的FCI，以及iii）通过密度矩阵重整化群（DMRG）获得的矩阵乘积状态表示。我们发现，DMRG一般给出了最有效的波函数表示的所有系统，特别是在1D链与本地化的基础。对于2D和3D系统，所有方法都在离域基础上表现最好，并且sCI的效率更接近DMRG的效率，前者在强相关区域中具有大约两倍的精度体积。与sCI相比，通常发现SVD-FCI方案需要稍微更多的参数来实现相同的能量精度。
英文摘要：
This work explores the ability of classical electronic structure methods to efficiently represent (compress) the information content of full configuration interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen model systems of different dimensionality and distinctive electronic structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D close-packed pyramid. To assess the ability of a computational method to produce accurate and compact wave functions, we introduce the accuracy volume, a metric that measures the number of variational parameters necessary to achieve a target energy error. Using this metric and the hydrogen models, we examine the performance of three classical deterministic methods: i) selected configuration interaction (sCI) realized both via an a posteriori and variational selection of the most important determinants, ii) rank-reduced FCI, obtained by an a posteriori singular value decomposition of the FCI tensor (SVD-FCI), and iii) the matrix product state representation obtained via the density matrix renormalization group (DMRG). We find that DMRG generally gives the most efficient wave function representation for all systems, particularly in the 1D chain with a localized basis. For the 2D and 3D systems, all methods perform best with a delocalized basis, and the efficiency of sCI is closer to that of DMRG, with the former having and accuracy volume approximately twice as large in the strong correlation regime. Compared to sCI, the SVD-FCI scheme is generally found to require a slightly larger number of parameters to achieve the same energy accuracy.
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