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Helium states II. Perturbation treatment of the ground state using the coordinates r < and r >
Author(s) -
Ramaker D. E.,
Hagnauer G. L.,
Carlson R. E.,
Gaetano R. F.,
Crain H.,
Schrader David M.
Publication year - 1968
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560020408
Subject(s) - eigenfunction , ground state , perturbation (astronomy) , hamiltonian (control theory) , wave function , helium , physics , helium atom , zeroth law of thermodynamics , quantum mechanics , mathematical physics , atomic physics , mathematics , eigenvalues and eigenvectors , mathematical optimization
Abstract 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.