Calculation of expectation values correct to second order applied to the ground state of the helium isoelectronic sequence

Abstract

A method is discussed for the determination of single particle atomic expectation values, (W), accurate to 0(δ2) if an approximate wave function, Ψ0T, correct to 0(δ) is employed. The technique involves writing the expectation values as a functional of Ψ0T, the Hamiltonian, and a subsidiary function which is the solution of a differential equation, derived from certain self-consistency considerations, involving both W and Ψ0T. Analytic calculations for , n = 2, 1, - 1, -2, and the electron density at the origin p(0) have been performed for the ground state of the helium isoelectronic sequence using energy-optimized simple products of hydrogenic states as the approximate wave function. For 1 ≤ Z ≥ 8, our results have an accuracy equivalent to a Hartree-Fock calculation when compared to both the results of Pekeris and a 204 term variational perturbation treatment. The least accurate of our results are for which for He and Li+ are in error by 1% and 0.3% respectively and in all cases this error is further diminished with each increase in the atomic number.