Relativistic Hirshfeld atoms in a molecule: An information-theoretic view, with application to Drude oscillator dispersion models

Abstract

Several ad hoc dispersion models for density-functional theory are based on the use of Hirshfeld (or "stockholder") partition of a molecular charge density, which provides an in situ definition of atomic size. We show that a recently introduced "optimized" quantum Drude oscillator model for dispersion admits a closed-form solution in terms of the Lambert $W$ function, whose branches identify the compact and diffuse oscillator solutions. The compact solution determines the $C_8$ (dipole-quadrupole) dispersion coefficient analytically from the free-atom polarizability, $C_6$ coefficient, and van der Waals radius, without any reference $C_8$ data. Next, we provide a formal basis for a relativistic version of the atoms-in-molecule Hirshfeld partition. Using four-component Dirac-Hartree-Fock densities for isolated atoms defines a strictly positive deformation field that carries the relativistic changes in atomic density into the Hirshfeld partition. A uniqueness theorem for the non-relativistic case is extended to relativistic Hirshfeld atoms and admits an asymptotic expansion through quadratic order in the fine-structure constant. Normalization requires the relativistic density correction to reshape the reference atom while preserving its population. Finally, four-component polarizabilities and C6 coefficients are reported for closed-shell atoms and ions, which supply the reference data required to extend atoms-in-molecules dispersion models into the heavy-element regime. Periodic trends are observable in a scalar contraction factor that measures relativistic effects.