非对角低秩性质：低标度计算化学方法的新机遇
The off-diagonal low rank property: new opportunities for low-scaling computational chemistry methods
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中文总结 AI 辅助
本文指出计算化学中多种矩阵具非对角低秩（ODLR）性质，综述其数学应用并首次证明福克矩阵等满足该性质，为无隙体系线性标度电子结构计算开辟新路径。
中文摘要 AI 辅助
计算化学中的许多矩阵既非稀疏也非低秩，这使得低标度算法的设计十分困难。在这篇观点文章中，我们指出计算化学中的诸多重要矩阵，如库仑矩阵、电子排斥积分张量、密度矩阵、定域分子轨道（LMO）系数矩阵、福克矩阵及核 Hessian 矩阵，均具有同一性质：当基函数被适当排序后，它们的非对角块具有低数值秩（尽管整体数值秩较高）。这一被称为非对角低秩（ODLR）的性质已在数学领域得到广泛研究，但在计算化学领域的应用却少得惊人。本文综述了现有数学文献中利用矩阵 ODLR 性质进行紧凑存储及高效计算或使用的方法，随后梳理了 ODLR 性质在计算化学中的应用现状，并指出了利用 ODLR 性质为稠密满秩矩阵设计新型低标度方法的潜在未来机遇。特别地，我们首次证明了福克矩阵和 LMO 系数矩阵即使在体系无隙（此时矩阵为稠密）的情况下也满足 ODLR 性质，这为零电子温度下无隙体系的线性标度电子结构计算铺平了道路。
英文摘要
Many matrices in computational chemistry are neither sparse nor low-rank, making the design of low-scaling algorithms difficult. In this Perspective, we point out that many important matrices in computational chemistry, such as the Coulomb matrix, the electronic repulsion integral tensor, the density matrix, the localized molecular orbital (LMO) coefficient matrix, the Fock matrix, and the nuclear Hessian matrix share the same property: when their basis functions are suitably ordered, their off-diagonal blocks have low numerical ranks (despite that they as a whole have high numerical ranks). This property, termed off-diagonal low rank (ODLR), has been extensively studied in the mathematics community, but has surprisingly found very little use in computational chemistry. This Perspective reviews the existing mathematical literature on how to use the ODLR property of matrices to compactly store, as well as efficiently calculate or use them. Subsequently, we review the use of the ODLR property in computational chemistry, and point out possible future opportunities of devising new low-scaling methods for dense, full-rank matrices, exploiting the ODLR property. In particular, we prove for the first time that Fock matrices and LMO coefficient matrices satisfy the ODLR property, even if the system is gapless (in which case the matrices are dense). This paves the way to linear scaling electronic structure calculations of gapless systems at zero electronic temperature.
