Shannon and Rényi Entropies of Molecular Densities Under Atomic Partitioning: Insights Into Extensity and an Incomplete Description of Electron Correlation

ABSTRACT In this work, we investigate the reliability of information‐theoretic measures based on the electron‐density and shape‐function, specifically Shannon and Rényi entropies, as descriptors of electronic correlation. By establishing a rigorous decomposition of these entropic measures into additive and nonadditive contributions, supported on a Mulliken‐like atomic partition of molecules, we systematically analyze the asymptotic behavior of the entropies at the infinite‐internuclear‐distance limit to assess the problem of static correlation and extensivity. Our algebraic and numerical analysis reveals several flaws in the use of these density‐based descriptors. We demonstrate that for minimal‐basis and different theoretical levels, the Shannon and Rényi entropies fail to encode the amount of static correlation conveyed by the underlying wavefunction. Conversely, shape‐function Shannon entropies and Rényi entropies (for ) violate extensivity. In larger basis sets, uncorrelated Hartree‐Fock densities consistently overestimate entropy compared to sufficiently correlated (e.g., full‐valence‐CAS) densities. Moreover, the entropies for insufficiently correlated methods violate extensivity. These findings indicate that electron‐density‐based measures are insufficient for capturing static correlation, suggesting that robust entropic descriptors should be constructed from higher‐dimensional Hilbert‐space objects.