量子化学中的张量数值方法：从Hartree-Fock能量到激发态
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
AI总结：
本文探讨了基于网格的张量数值方法在实空间电子结构计算中的应用，提出了一种基于低秩表示的3D Hartree-Fock本征值求解器，并介绍了张量方法在MP2能量修正和Bethe-Salpeter激发态计算中的应用。
摘要：
我们回顾了基于网格的张量数值方法最近的成功，并讨论了其在实空间电子结构计算中的前景。这些方法基于多维函数和积分算子的低秩表示，导致完全基于网格的张量结构3D Hartree-Fock本征值求解器。它受益于使用基于基函数、电子密度和卷积积分算子的低秩近似，以O(n log n)复杂度计算核心哈密顿量和两个电子积分（TEI）。TEI张量的计算算法基于多重分解，使用代数1D“密度拟合”方案。基函数不限于分离的高斯函数，因为解析积分被高精度张量结构数值积分取代。最近引入了基于低秩分解和缩减基方法的张量方法用于后Hartree-Fock计算，包括MP2能量修正和Bethe-Salpeter激发态计算。另一个方向是开发基于张量的有限晶格结构系统的Hartree-Fock数值方案，其中一个数值挑战是求和大量原子核的静电势。3D网格基于张量方法计算L×L×L晶格上的势能和，表现出线性于L的计算工作O(L)，而不是通常的O(L^3 log L)标量方法。
英文摘要：
We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n $ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches.
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