Helium states II. Perturbation treatment of the ground state using the coordinates r < and r >

In an attempt to improve upon the convergence properties of the Hylleraas‐Scherr‐Knight‐Midtdal perturbation expansion for the ground‐state energies and eigenfunctions of the helium isoelectronic sequence, the term r   > −1is included in the zeroth‐order Hamil‐tonian. This term dominates the usual perturbation r   12 −1for the ground state of these systems, and by removing it from H (1) we substantially reduce, in some sense, its size. In order to find the exact eigenfunction of the resulting zeroth‐order Hamiltonian it was found necessary to include in H (0) two additional terms involving the delta function δ( r 1 − r 2 ) = δ( r < − r > ) and one such term in H (1) . Approximate first‐ and second‐order eigenfunctions are calculated variationally giving the energies to fifth order. The results are disappointing. The errors in the energies to fifth order for He, Li + , and Be 2+ , although quite small, are significantly larger than the corresponding errors in the more conventional perturbation treatment. Reasons for the failure to improve upon the earlier results are discussed. A “paradox” noted some time ago by Snyder and Parr is examined in an Appendix.