Continuous Information Descriptors for Electron Localization: Relativistic Spatial Responses, Nonadditivity, and...

Accurately simulating heavy elements requires resolving spatial electron density competitions often obscured by traditional scalar energy metrics. Using continuous information-theoretic descriptors, we evaluate the spin-free exact two-component (sf-X2C) Hamiltonian and exact-exchange interactions across the periodic table (Z = 1-86). From the analytical hydrogenic baseline to generalized many-body frameworks, we demonstrate how the nonlinear radial decay of the effective nuclear charge dictates the macroscopic competition between direct core contraction and indirect d/f-block spatial expansion. We reveal the intrinsic spatial nonadditivity of these logarithmic descriptors. Unlike scalar energies, the coupling between relativistic kinematics and exact exchange generates state-dependent spatial cross-terms driven by localized d/f electrons, necessitating fully coupled treatments for heavy elements. Extending to molecular architectures, we introduce the continuous Kullback-Leibler (KL) divergence against a promolecular reference to quantify the spatial information cost of chemical bonding. This geometric probe decouples the concentric spatial contraction of main-group covalent bonds from the anisotropic lobe-like expansion along transition metal coordination axes. These spatial divergence signatures offer three-dimensional diagnostic constraints for parametrizing next-generation heavy-element density functional approximations.