Updated all-electron Dirac--Fock densities and an element-adaptive parameterisation of scattering factors and potentials for neutral atoms

Abstract

Updated reference data and an analytic parameterisation of elastic electron and X-ray scattering are presented for all 118 neutral atoms. The reference electron densities for the multi-electron elements $Z=2$--$118$ are computed with the relativistic B-spline Dirac--Fock code atomx, while hydrogen is constructed from the exact relativistic one-electron Dirac $1s$ solution; the electron scattering factor $f_e(g)$, X-ray scattering factor $f_x(g)$ and radial moments are derived from these densities. The reported parameterisation extends the fixed-size Lobato--Van Dyck hydrogenic expansion while retaining closed-form expressions for $f_x(g)$, $ρ(r)$, the electrostatic potential $V(r)$ and the projected potential $V(R)$. These extensions are an element-adaptive basis size $n_t(Z)$, a simultaneous real- and reciprocal-space fit, an exact $\langle r^4\rangle$ constraint in place of the non-relativistic Kato cusp, and a charge-carrying Dirac--Padé basis term that adds a polynomial-times-exponential shape channel without replacing the hydrogenic basis by a tabulated Dirac radial function or assigning the term to a physical shell. For the same total parameter count, the Dirac--Padé-enriched basis improves on the parameter-matched non-relativistic basis for 111 of the 118 elements, lowering the mean total cost by 39\%. Relative to a controlled fixed five-term refit on the same reference grid and objective, the element-adaptive bases improve the median reciprocal-space deviations by about three to four orders of magnitude and resolve shell structure in $4πr^2ρ(r)$ that the fixed five-term basis cannot. The largest changes occur near the nucleus and in the reciprocal-space tail beyond the legacy 12 inverse angstroms range, which is directly relevant to quantitative high-angle scattering and electron-diffraction measurements.