Closed-form evaluation of the cyclic four-electron atomic correlation integral
Abstract
For positive exponents α1,…,α4 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[Li3(1/4)/2−7ζ(3)/24+π2log 2/12−2log3 2/3], a weight-three period over 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.
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BibTeX
@misc{matuzas2026closedform,
title = {{Closed-form evaluation of the cyclic four-electron atomic correlation integral}},
author = {Jonas Matuzas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00036},
eprint = {2610.00036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00036},
}
RIS
TY - UNPB
TI - Closed-form evaluation of the cyclic four-electron atomic correlation integral
AU - Jonas Matuzas
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00036
UR - https://arxiv.org/abs/2610.00036
AB - For positive exponents α_1,…,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[Li_3(1/4)/2−7ζ(3)/24+π^2log 2/12−2log^3 2/3], a weight-three period over 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.
ER -