来自实部和复部固体谐波积分引擎的结构化高角动量库仑张量：一种视角
Structured High-Angular-Momentum Coulomb Tensors from Real and Complex Solid-Harmonic Integral Engines: A Perspective
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中文总结 AI 辅助
研究电子排斥积分相关问题，通过直接实部或复部固体谐波引擎在目标空间工作，产生更小库仑张量并保持相关标签，解析计数指导多项计算，为高效积分生成与多电子计算搭建桥梁。
中文摘要 AI 辅助
电子排斥积分描述了由轨道基函数构建的电荷分布之间的库仑相互作用。大多数积分算法通过笛卡尔高斯函数生成这些量，其角形状写成\(x\)、\(y\)和\(z\)的幂，然后将结果转换为球函数。这种方法有效，但从\(d\)壳层起，笛卡尔表示包含比计算所需球空间更多的函数。直接实部或复部固体谐波引擎从一开始就在目标空间中工作。因此，它们在保持描述其角结构的顺序、相位和磁量子数标签的同时，产生更小的最终库仑张量。超越积分评估遵循这种结构揭示了与使用该张量的算法的直接联系。简单的解析计数量化了张量大小、角块、径向斯莱特 - 康登参数和对空间工作。这些量指导低秩分解、局部哈密顿量构建、量子模拟以及到旋量或有效模型基的变换。通过这种方式，固体谐波积分引擎在高效积分生成和结构化多电子计算之间提供了直接桥梁。
英文摘要
Electron-repulsion integrals describe the Coulomb interaction between charge distributions built from orbital basis functions. Most integral algorithms generate these quantities through Cartesian Gaussian functions, whose angular shapes are written as powers of $x$, $y$, and $z$, and then transform the result to spherical functions. This route is effective, but from $d$ shells onward the Cartesian representation contains more functions than the spherical space required by the calculation. Direct real or complex solid-harmonic engines work in that target space from the beginning. They therefore produce a smaller final Coulomb tensor while preserving the ordering, phase, and magnetic-quantum-number labels that describe its angular structure. Following this structure beyond integral evaluation reveals direct connections to the algorithms that use the tensor. Simple analytical counts quantify tensor size, angular blocks, radial Slater--Condon parameters, and pair-space work. These quantities guide low-rank factorization, local Hamiltonian construction, quantum simulation, and transformations to spinor or effective-model bases. In this way, solid-harmonic integral engines provide a direct bridge between efficient integral generation and structured many-electron computation.
