Grid-free quantum crystallographic refinement using unbiased analytic atomic form factors from density fitting

Abstract Accurate crystal structure determination requires reliable models of how atoms scatter X-rays and electrons. While quantum crystallographic methods such as Hirshfeld atom refinement (HAR) markedly improve upon the widely used Independent Atom Model (IAM) by incorporating ab initio electron densities, their reliance on numerical real-space grids imposes a significant computational bottleneck. Here, we present a density-fitting-based refinement scheme that addresses this limitation by expanding the molecular electron density via the Resolution-of-the-Identity (RI) approach into atom-centred Gaussian contributions. This enables fully analytic evaluation of aspherical X-ray and electron scattering factors, eliminating the need for real-space grids or reference atoms entirely. Benchmark refinements on a diverse set of small-molecule crystals within Olex2/NoSpherA2 demonstrate that the RI-based model reproduces R values, residual electron density, X–H bond lengths, and anisotropic displacement parameters on par with HAR, while offering substantially improved computational efficiency. This is most prominent particularly at high diffraction resolution and for larger asymmetric units.