A Bi−directional method for evaluating integrals involving higher transcendental functions. HyperRAF: A Julia package for new hyper−radial functions

The electron repulsion integrals over Slater−type orbitals with non−integer principal quantum numbers are investigated. These integrals are useful in both non−relativistic and relativistic calculations of many−electron systems. They involve hyper−geometric functions that are practically difficult to compute. Relationships free from hyper−geometric functions for expectation values of Coulomb potential (r_{21}^{-1}) are derived. These relationships are new and show that the complication coming from two−range nature of Laplace expansion for the Coulomb potential is removed. This is achieved by utilizing auxiliary functions represented in finite power series. They serve as essential components in deriving straightforward recurrence relationships for electron repulsion integrals. In the context of computing the expectation values of potentials with arbitrary power, the methodology presented here for evaluation of these integrals forms the initial condition. It is also adapted to multi−center integrals.

本研究探讨了基于斯莱特型轨道的非整数量子数电子排斥积分。此类积分在非相对论及相对论的多电子系统计算中均具有重要价值。这些积分涉及超几何函数，其实际计算颇为繁难。本研究推导出免于超几何函数关系的库仑势（r_{21}^{-1}）期望值。这些新关系揭示了库仑势拉普拉斯展开双范围性质所引入的复杂性得以消除。这一成就得益于有限幂级数表示的辅助函数的运用，它们作为推导电子排斥积分直接递推关系的基石。在计算任意幂势的期望值背景下，本文提出的评估这些积分的方法构成了初始条件。该方法亦被推广应用于多中心积分。