Results of an analytic calculation for the coherent atomic scattering factor for the ground state of the helium isoelectronic sequence

The analytical method developed by the authors for the determination of the expectation value of single-particle operators W = Σ_i W(r_i) correct to second order is employed to obtain an analytic expression for the coherent atomic scattering factor F(k) for the ground state of the helium isoelectronic sequence valid for all values of momentum transfer. The trial wave function ϕ_ot employed is an energy minimized Hartree product of hydrogenic states. For helium the results for the form factor are within 1.2% of the highly accurate values calculated using a 120 parameter configuration interaction wave function and have an accuracy equivalent to that of an analytical Hartree-Fock treatment. This error is further reduced as the atomic number is increased. The calculations are extended to the infinite momentum transfer range and the accuracy of the results are studied in this limit by employing the Kato electron-nucleus cusp condition for the exact ground state wave function of two-electron atomic systems. We observe that in the infinite momentum transfer limit our results for helium are in error by 0.57% and that this error is further diminished for each successive element of the isoelectronic sequence. In addition, we note that the cusp condition is exactly satisfied and is independent of the variational parameter employed in the trial wave function.