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.

Automated summary

The electron repulsion integrals over Slater-type orbitals with non-integer principal quantum numbers are investigated in this study. These integrals are important in calculations of many-electron systems. New relationships free from hyper-geometric functions are derived to simplify the calculations. With the use of auxiliary functions and straightforward recurrence relationships, these integrals can be efficiently computed, providing initial conditions for the evaluation of expectation values and potentials.