A Hartree–Fock approach to the Steklov eigenproblem for a two-electron atom in an s² state

Abstract

Two decades ago, in a brief note (Hinze and Hamacher, J Chem Phys 92:4372–4373, 1990), Hinze and Hamacher made a conjecture that the Hartree–Fock method may be applied to determine approximate solutions to a non-standard (Steklov-type) eigenproblem encountered in the non-relativistic eigenchannel R-matrix method. Later, this thread was mathematically pursued further by the present author (Szmytkowski, Phys Rev A 61:022725, 2000; erratum 66:029901, 2002). In the present paper, which is dedicated to the memory of Professor Jürgen Hinze, we make an attempt to elucidate the idea underlying the aforementioned works. To focus on the essence and avoid obscuring mathematical details, we consider the simplest system which is a two-electron atom in an s² state. Variational principles for pertinent Steklov eigenvalues (i.e., eigenvalues of a two-electron Dirichlet-to-Neumann integral operator) and for their reciprocals (i.e., eigenvalues of a two-electron Neumann-to-Dirichlet integral operator) are used to derive a radial integro-differential eigensystem of a Hartree–Fock type, in which both a Lagrange multiplier in an integro-differential equation and a Steklov eigenvalue appearing in a boundary condition are to be determined simultaneously at a fixed total energy of the atom. Mathematical similarities and (particularly) differences between the problem considered here and that of a spherically confined two-electron atom are highlighted.