Shannon and R\'enyi entropies of molecular densities: insights into extensivity and the incomplete description of electron correlation

TL;DR

This study critically assesses Shannon and R\'enyi entropies derived from electron densities, revealing their limitations in accurately describing static electron correlation and extensivity in molecular systems.

Contribution

It provides a rigorous decomposition of density-based entropies and demonstrates their failure to reliably encode static correlation across different theoretical levels.

Findings

Shannon and R\'enyi entropies fail to encode static correlation accurately.

Shape-function entropies violate extensivity for \(\alpha \neq 1\).

Uncorrelated Hartree-Fock densities overestimate entropy compared to correlated densities.

Abstract

In this work, we investigate the reliability of information-theoretic measures based on the electron-density and shape-function, specifically Shannon and R\'enyi 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 R\'enyi entropies fail to encode the amount of static correlation conveyed by the underlying wavefunction. Conversely, shape-function Shannon entropies and R\'enyi entropies (for $\alpha \neq 1$) 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 constructed from higher-dimensional Hilbert-space objects.