Singular Hylleraas three-electron integrals

TL;DR

This paper develops recursion relations for singular three-electron integrals in Hylleraas basis, enabling high-precision calculations of quantum electrodynamics effects in few-electron systems.

Contribution

It introduces new recursion relations and methods for evaluating singular Hylleraas integrals, advancing computational techniques in quantum physics.

Findings

Derived recursion relations for singular integrals.

Enabled high-precision numerical evaluations.

Facilitated calculations of QED effects in few-electron systems.

Abstract

Calculations of the leading quantum electrodynamics effects in few electron systems involve singular matrix elements of the inter-electronic distances of the form $1/r_i^3$ and $1/r_{ij}^3$. Integrals that result when the nonrelativistic wave function is represented with a Hylleraas basis representation are studied. Recursion relations for various powers of the electron coordinates and the master integrals are derived in a form suited for high precision numerical evaluations.