The Angular Localization Function (ALF): a practical tool to measure solvent angular order with Molecular Density Functional Theory

Molecular density functional theory is a powerful technique for efficiently computing the spatially and orientationally dependent equilibrium density of a molecular solvent around an arbitrary solute. This density encodes the detailed solvent structure, but contains so much information that it is difficult to interpret comprehensively. Although spatial dependence is frequently analyzed through orientationally integrated number density, angular information remains poorly exploited. The present work addresses this gap by introducing a function that provides a local measure of the angular order: the Angular Localization Function (ALF), derived from the ideal free energy functional, which quantifies the entropy. We discuss the connections between ALF and well known statistical functions. We illustrate the utility of ALF by discussing the solvent structure for three systems immersed in water: water as a solute, an octanol molecule, and three clay minerals (talc, fluorotalc and pyrophyllite) with small differences in their structure leading to subtle effects on their interactions with water. ALF provides information complementary to quantities such as the average polarization or charge density to characterize the local orientational distribution of solvent molecules around solutes and next to surfaces. It also offers a convenient visualization tool akin to the Electronic Localization Function (ELF) used to analyze bonding in quantum chemistry.

💡 Research Summary

The authors introduce the Angular Localization Function (ALF), a novel scalar field designed to quantify the local orientational order of a molecular solvent as obtained from Molecular Density Functional Theory (MDFT). MDFT provides the equilibrium solvent density ρ(r,Ω) that depends on three spatial coordinates and three Euler angles, but visualizing and interpreting this six‑dimensional object is notoriously difficult. Existing analyses usually collapse the data onto lower‑dimensional representations such as number density n(r), radial distribution functions, or the first angular moment (the polarization field). These projections discard most of the angular information, leaving a gap in our understanding of how solvent molecules are oriented around solutes or interfaces.

The authors start from the ideal‑gas contribution to the MDFT free‑energy functional, F_id