Fortuitous Error Attenuation in Hirshfeld Charges from Gaussian-Based Spherical Atomic Densities

Spherical atomic electron densities are central to the construction of promolecular densities and Hirshfeld stockholder weights. In this work, spherical densities generated from WFX wave function files using Gaussian-type orbital (GTO) basis sets are systematically compared with the reference Slater-type orbital (STO) atomic densities reported by Bunge and co-workers. Hartree-Fock wave functions expanded in STO-3G, 6-31G*, 6-311G*, def2TZVP, and cc-pVTZ basis sets were considered for atoms from He to Kr, and the ANO-pVTZ basis set also was tested from He to Ar. The results show that flexible Gaussian basis sets reproduce the overall radial behavior of the reference densities, but significant discrepancies remain in the near-nuclear and asymptotic regions. These differences become much more apparent when derivative-based quantities are examined, particularly through |∇ρ(r)|and the logarithmic derivative |∇ρ(r)|/ρ(r), which expose the intrinsic limitations of Gaussian expansions in describing the electron-nucleus cusp and the long-range density decay. The analysis was extended to molecular systems by constructing promolecular densities from both STO and GTO-derived spherical atomic densities. Difference maps and L1 norms show that atomic-level discrepancies are transferred to promolecular densities and to the corresponding Hirshfeld partition weights. The weight differences, defined as δwA(r) = wSTOA(r) - wGTOA(r), show that the choice of atomic reference density affects not only the promolecular density, but also the spatial distribution of the stockholder weights. However, when Hirshfeld populations are evaluated, the effect of these local differences is strongly attenuated because the weight differences enter the population integral after multiplication by the molecular electron density. Accordingly, the population differences ΔNA = NSTOA - NGTOA are generally small for sufficiently flexible basis sets, even when the corresponding weights exhibit sizable spatial deviations in low-density regions. These findings show that Gaussian-based spherical densities can provide reliable Hirshfeld populations and charges when adequate basis sets are used, whereas density derivatives, promolecular densities, and local partition weights remain more sensitive probes of the quality of the underlying atomic reference densities.