Related Experiment Video
Updated: Aug 23, 2026

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
Published on: April 8, 2020
Highly accurate evaluation of atomic three-electron integrals of lowest orders
Frank E Harris1, Alexei M Frolov, Vedene H Smith
1Department of Physics, University of Utah, Salt Lake City, UT 84112, USA. harris@qtp.ufl.edu
Abstract:
Calculations of three-electron atomic systems in Hylleraas coordinates require integrals involving all the interparticle distances r(ij), which have usually been evaluated by introducing series expansions. For integrals with the smallest powers of r(ij) these expansions do not converge at a satisfactory rate, leading some investigators to introduce convergence-acceleration procedures. This paper recommends the alternative of evaluating these integrals in closed form and presents stable explicit formulas for so doing. Some of the formulas are more compact versions of those in the literature; others have not been previously reported. It is also shown that finite-difference methods can be used with advantage to obtain additional low-order integrals. Sample integral values have been provided for test purposes.
Related Concept Videos
Electronic Structure of Atoms
An atom comprises protons and neutrons, which are contained inside the dense, central core called the nucleus, with electrons present around the nucleus. Taking into account the wave–particle duality of electrons and the uncertainty in position around the nucleus, quantum mechanics provides a more accurate model for the atomic structure. It describes atomic orbitals as the regions around the nucleus where electrons of discrete energy exist, characterized by four quantum numbers: n, l, ml, and...
Atomic Orbitals
The Quantum-Mechanical Model of an Atom
Electron Orbital Model
Atomic Radii and Effective Nuclear Charge
The Energies of Atomic Orbitals

