Nodal surfaces of helium atom eigenfunctions

Using a rapidly convergent composite basis of Frankowski-Pekeris and Frankowski functions, we have accurately calculated the nodal surfaces of low-lying excited states of the helium atom to investigate Bressanini and Reynolds’ conjecture [D. Bressanini and P. J. Reynolds, Phys. Rev. Lett. 95, 110201 (2005)] that these nodal surfaces are rigorously independent of the interelectronic angle θ₁₂ . We find that in fact there is a slight dependence of the nodal surfaces on θ₁₂ , but it is so small that the assumption of strict independence may well yield extremely useful approximations in fixed-node quantum Monte Carlo calculations. We explain how Kato’s cusp conditions determine the qualitative features of these nodal surfaces, which can accurately be modeled using the familiar ansatz of a symmetric or antisymmetric linear combination of products of hydrogenic orbitals, with some adjustments of the parameters. We explain why a similar near independence of the nodal surfaces on the angular variables can be expected for the ground and singly excited states of the lithium atom, but generally not for larger atoms.