How do interacting quantum atoms energy components depend on the choice of partitioning method? Disentangling the effects of atomic weights and populations

Interacting Quantum Atoms (IQA) is a real-space energy decomposition framework that reveals how individual atoms and their pairwise interactions contribute to the stabilization of molecules, yet its application has been dominated by the Quantum Theory of Atoms in Molecules (QTAIM). Although fuzzy partitioning schemes offer smoother atomic densities, reduce numerical costs, and avoid complications associated with non-nuclear attractors, their influence on IQA energy components has not been systematically assessed. We investigate how the choice of atomic partitioning affects IQA energy components by comparing QTAIM with several variational Hirshfeld-based schemes. To enable direct comparisons between QTAIM- and fuzzy-partitioned IQA energy components, we integrate charge-constrained variational Hirshfeld partitioning with the IQA framework, generating fuzzy analogs of QTAIM atoms with identical atomic charges. Across a diverse set of diatomic and polyatomic molecules, we observe that fuzzy partitioning consistently results in less negative intra-atomic energies and more stabilizing interatomic interactions than QTAIM. These differences are traced to changes in the underlying electron-nuclear attraction and electron-electron repulsion terms. The proposed methodology provides a rigorous basis for comparing QTAIM and fuzzy atoms in IQA, enabling the effects of atomic populations and partitioning schemes to be disentangled for the first time.