循环四电子原子关联积分的闭式求值
Closed-form evaluation of the cyclic four-electron atomic correlation integral
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中文总结 AI 辅助
本文针对循环四电子 Slater 积分（Bonham 分类情形 V）给出闭式求值，以十八项对数-多对数表达式表示，并证明其对称性与极限定律，为四电子关联计算提供精确基准。
中文摘要 AI 辅助
对于正指数 $\alpha_1,\ldots,\alpha_4$，我们以有限实表达式（权重至多为三）求值循环四电子 Slater 积分，其拓扑记录在 Bonham 1965 年分类中的情形 V。一般答案是一个显式的十八项和，仅涉及指数的有理函数的对数、双对数函数和三对数函数；所有对数自变量均为正，所有非常数多对数自变量可置于 $(-1,1)$ 内，因此无需根式、广义多对数、围道处方或未求值极限。在相等指数下，该值坍缩为 $I(\lambda,\lambda,\lambda,\lambda)=(4\pi)^4\lambda^{-4}[\mathrm{Li}_3(1/4)/2-7\zeta(3)/24+\pi^2\log 2/12-2\log^3 2/3]$，这是一个仅在 2 和 3 处分歧的 $\mathbb{Q}$ 上的权重三周期。三条独立实现的数值路径在四个指数元组上一致至至少 32 位小数，经典公式的精确二进求值给出八个 100 位证书，三个已发表的四电子校准值被复现至每个印刷数字。我们还证明了小指数和大指数极限定律（具有闭式系数）、二面体对称性，以及四个指数的三种不等价循环放置的严格重排定律。该公式适用于四环，不评估两种弦化拓扑。
英文摘要
For positive exponents $α_1,\ldots,α_4$ we evaluate the cyclic four-electron Slater integral, the topology recorded as case V in Bonham's 1965 classification, in a finite real expression of weight at most three. The general answer is an explicit sum of eighteen blocks involving only logarithms, dilogarithms and trilogarithms of rational functions of the exponents; all logarithm arguments are positive and all nonconstant polylogarithm arguments may be placed in $(-1,1)$, so no radical, generalized polylogarithm, contour prescription or unevaluated limit is required. At equal exponents the value collapses to $I(λ,λ,λ,λ)=(4π)^4λ^{-4}[\mathrm{Li}_3(1/4)/2-7ζ(3)/24+π^2\log 2/12-2\log^3 2/3]$, a weight-three period over $\mathbb{Q}$ ramified only at 2 and 3. Three independently implemented numerical routes agree at four exponent tuples to at least 32 decimal places, an exact-dyadic evaluation of the classical formula gives eight 100-place certificates, and three published four-electron calibration values are reproduced to every printed digit. We also prove the small- and large-exponent limit laws with closed-form coefficients, dihedral symmetry, and a strict rearrangement law for the three inequivalent cyclic placements of four exponents. The formula applies to the four-cycle and does not evaluate the two chorded topologies.
