Analytic Calculation of the Coherent Atomic Scattering Factor for the Ground State of the Helium Isoelectronic Sequence

Abstract

The analytical method developed by the authors for the determination of the expectation value of single-particle operators $W={\ensuremath{\Sigma}}_{i}{W(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i})$ correct to second order is employed in both the first and second decoupling approximations to obtain analytic expressions for the coherent x-ray scattering factor $F(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ for the ground state of the helium isoelectronic sequence valid for all values of momentum transfer. The trial wave function ${\ensuremath{\psi}}_{0T}$ employed is an energy-minimized Hartree product of hydrogenic states. For helium the results for the form factor in the second decoupling approximation are found to be superior to those of the first, and 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. In order to demonstrate the internal self-consistency of the technique, we prove that the expectation value of any single-particle operator as obtained by direct use of the analytical method is the same as the expectation value obtained employing the form factor provided that the latter has also been calculated by the analytical method employing the same trial wave function. Finally, we extend our calculations to the infinite-momentum-transfer range and study our results in this limit by employing a cusp condition for the exact ground-state wave function of two-electron atomic systems written in terms of the logarithmic derivative of the electron density at the origin. 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 via the formalism of the second decoupling approximation and is independent of the variational parameter employed.