, Volume 68, Issue 6, pp 445-469

r 12-Dependent terms in the wave function as closed sums of partial wave amplitudes for large l

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Abstract

The ansatz Ψ= (1+1/2r12)Φ+χ with Φ the bare nuclear (or screened nuclear) wave function and χ expanded in products of one-electron functions is explored for second-order perturbation theory and for variational calculations of the ground state of Helium-like ions.

The energy increments E l (2) corresponding to the partial wave expansion of χ go asymptotically as l−8, while conventional partial wave increments go as l−4. χ is coupled to Φ by a “residual” interaction U12 that has no singularity for r12=0. With the present ansatz it is sufficient to include l-values up to 5 in order to get the second-order energy accurate to one microhartree. For the same accuracy l≤4 is sufficient in a “CI with correlated reference function” while in conventional CI one must go to l∼50. The surprisingly faster convergence of the variational approach as compared to second-order perturbation theory is explained. The slow convergence of the traditional partial wave expansion is entirely due to the attempt to represent the quantity 1=〈Φ¦r12r12 −1¦Φ〉 by its partial wave expansion. The best reference function Φ shows very little shielding and resembles closely the eigenstate of the bare nuclear Hamiltonian. The generalization to arbitrary systems is discussed and it is pointed out that the calculation of “difficult” integrals can be avoided without a significant loss in accuracy.