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 12. We find that in fact there is a slight
dependence of the nodal surfaces on 12, 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.