Extraction of directional electron-density features from diffraction data using spherical-harmonic decomposition

Abstract

Directional anisotropy in electron density provides key information about chemical bonding that is not readily accessible from conventional electron-density maps. Here, a model-independent framework is presented for decomposing experimental structure factors into angular components using spherical harmonics. Reciprocal-space projection onto spherical harmonics followed by standard Fourier synthesis yields angularly filtered density maps. The{ell} = 0 component captures the isotropic part of the density, while the{ell} = 1 components resemble px, py and pz-like dipolar functions that highlight directional electronic structure. Applications to high-resolution datasets, including urea, the Gly-Ala dipeptide and a 0.97 [A]{beta}-lactamase structure, reveal chemically interpretable dipolar features associated with carbonyl and amide bonds, N-H interactions and aromatic{pi} systems. Quantitative analysis using bond-centred sampling demonstrates stable dipolar signatures that remain detectable under moderate resolution truncation. These results establish spherical-harmonic angular decomposition as a practical framework for extracting directional electronic information from crystallographic electron-density maps. SynopsisAngular decomposition of experimental structure factors reveals dipolar anisotropy and directional electron-density features that are directly meaningful for chemical interpretation.