Wigner intracule for the Kellner helium-like ions

The Kellner wavefunction for a helium-like ion is the Hartree–Fock solution wherein the orbital is a Slater-type function with the variationally optimal exponent ζ = Z − 5/16. The Wigner intracule W(u, v) of a system gives the joint quasiprobability of finding two electrons whose position space and momentum space separations are |r₁ − r₂| = u and |p₁ − p₂| = v, respectively. In this article, we extend Wigner intracule theory beyond Gaussian-type functions by deriving W(u, v) for the Kellner helium-like ions. Although we have not been able to express W(u, v) in closed form, our formulation reduces it to a two-dimensional integral that can be treated by quadrature. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004