Shannon and Rényi entropies of molecular densities under atomic partition: insights into extensivity and an incomplete description of correlation

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 α 1) 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