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arxiv: 2604.18081 · v1 · pith:7OGSCD5Lnew · submitted 2026-04-20 · 🪐 quant-ph · physics.chem-ph

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

Diogo J. L. Rodrigues , Evelio Francisco , \'Angel Mart\'in Pend\'as This is my paper

Pith reviewed 2026-05-10 05:19 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-ph
keywords Shannon entropyRényi entropyelectron densitystatic correlationextensivitymolecular dissociationquantum chemistry
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The pith

Electron-density entropies fail to capture static correlation and often violate extensivity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether Shannon and Rényi entropies derived from electron densities can serve as good measures of electronic correlation in molecules. By breaking these entropies into parts tied to individual atoms and looking at what happens when molecules are pulled apart to infinity, it finds that these density measures do not reflect the static correlation present in the wavefunction. Shape-function versions also fail to be extensive, meaning they do not scale properly with system size. This matters because it questions the use of simple density-based information measures for understanding correlation effects in chemistry.

Core claim

Through algebraic and numerical analysis, the paper shows that Shannon and Rényi entropies based on electron densities fail to encode the amount of static correlation in the wavefunction for minimal-basis sets and various theoretical levels. Shape-function Shannon entropies and Rényi entropies with alpha not equal to 1 violate extensivity. In larger basis sets, Hartree-Fock densities overestimate entropy compared to correlated densities, and insufficiently correlated methods violate extensivity. This indicates that electron-density-based measures are insufficient for capturing static correlation.

What carries the argument

The decomposition of entropic measures into additive and nonadditive contributions using a Mulliken-like atomic partition, combined with asymptotic analysis at the infinite-internuclear-distance limit.

If this is right

  • Shannon and Rényi entropies from densities do not match the static correlation in the underlying wavefunction.
  • Shape-function based Shannon and certain Rényi entropies violate extensivity.
  • Hartree-Fock densities overestimate entropy relative to correlated wavefunctions in larger basis sets.
  • Methods lacking sufficient correlation violate extensivity in their entropies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Robust entropic descriptors for correlation may require using higher-dimensional objects like the wavefunction or two-particle densities rather than one-particle density.
  • Information-theoretic tools in quantum chemistry might need redesign to properly handle dissociation limits and static correlation.
  • These findings could extend to other density-based functionals or information measures used in molecular analysis.

Load-bearing premise

The Mulliken-like atomic partition and the infinite-separation analysis accurately isolate and measure the static correlation and extensivity properties from the wavefunction.

What would settle it

If the difference in these entropies between uncorrelated and correlated calculations at dissociation does not align with known static correlation measures like the difference in energy or natural orbital occupations, that would challenge the claim of insufficiency.

Figures

Figures reproduced from arXiv: 2604.18081 by \'Angel Mart\'in Pend\'as, Diogo J. L. Rodrigues, Evelio Francisco.

Figure 1
Figure 1. Figure 1: Various entropic terms for H2 described with a minimal STO-6G basis for different methods at various internuclear distances (points). The sum of the free atomic entropies obtained with the same basis set is shown as a straight horizontal line. Note the convergence of all methods to solely net contributions that equal the isolated atom entropy values. The lowest R-value is the equilibrium internuclear dista… view at source ↗
Figure 2
Figure 2. Figure 2: Shape-function net and total entropies for STO-6G H2 as obtained from different methods at various internuclear distances (points). The entropy of the isolated atoms is depicted with a straight horizontal line. The lowest R-value is the equilibrium internuclear distance. Notice the non-extensivity of Sσ.). 2.3 The dissociation limit of the Shannon entropy Since the following expression for the density part… view at source ↗
Figure 3
Figure 3. Figure 3: Various entropic terms for N2 described with a minimal STO-6G basis for different methods at various internuclear distances (points). The sum of the free atomic entropies obtained with the same basis set is shown as a straight horizontal line. Note the convergence of all methods to solely net contributions that equal the isolated atom entropy values. The lowest R-value is the equilibrium internuclear dista… view at source ↗
Figure 4
Figure 4. Figure 4: Net and total entropies (shape-function based) terms for N2 at minimal-basis (STO-6G) for different methods at various internuclear distances (points) and entropy of the isolated atoms (lines). The lowest x-value is the equilibrium internuclear distance. Notice how the values at large distances are close in magnitude to the values of the H2 molecule, which is an advantage of using shape-function based entr… view at source ↗
Figure 5
Figure 5. Figure 5: H2/aug-cc-pVTZ at the HF and CAS(2,2) levels at different internuclear distances. Top: Sρ and S net ρ entropies. The sum of the entropies of the two H/aug-cc-pVTZ isolated atoms is shown as a horizontal line, and the lowest R-value is the equilibrium internuclear distance. Bottom: S overl ρ and −S nadd ρ Finally, a symmetric dissociation of a water molecule in which a simultaneous stretching of the two O-H… view at source ↗
Figure 6
Figure 6. Figure 6: N2/aug-cc-pVTZ at the HF and several CAS levels at different internuclear distances. Top: Sρ and S net ρ entropies. The sum of the entropies of the two ROHF/aug-cc-pVTZ isolated N atoms is shown as a horizontal line, and the lowest R-value is the equilibrium internuclear distance. Bottom: S overl ρ and −S nadd ρ . Only the full￾valence-CAS(10,10) calculation is size extensive and Sρ for the isolated N atom… view at source ↗
Figure 7
Figure 7. Figure 7: Total and net entropies for H2O at aug-cc-pVTZ for different methods at various internuclear distances (points) and entropy of the isolated atoms (lines). The internuclear distances are the O-H distance, which is simultaneously changed for both bonds, preserving the symmetry of the molecule during dissociation. The lowest x-value is the equilibrium internuclear distance. Notice how CAS(8,6) smoothly tends … view at source ↗
Figure 8
Figure 8. Figure 8: Overlap and nonadditive entropies for H2O at aug-cc-pVTZ for different methods at various internuclear distances (points) and entropy of the isolated atoms (lines). The lowest x-value is the equilibrium internuclear distance. Notice the vanishing of the overlap and nonadditive entropies for all methods. S overl = S overl,O + 2S overl,H = P A S overl,OA + P B S overl,HB. (We chose to plot the negative of th… view at source ↗
Figure 9
Figure 9. Figure 9: Net, nonadditive-intra (P A pA log pA) and total Rényi α = 2 entropy terms for H2 at minimal-basis (STO-6G) for different methods at various internuclear distances (points) and entropy of the isolated atoms (lines). The lowest R value is the equilibrium internuclear distance. Notice how the full entropy does not tend to the net entropy, but to the net entropy plus the nonadditive-intra entropy, as predicte… view at source ↗
Figure 10
Figure 10. Figure 10: Net, nonadditive-intra (P A pA log pA) and total Rényi α = 2 entropy terms for H2 at aug-cc-pVTZ basis for different methods at various internuclear distances (points) and entropy of the isolated atoms (lines). The lowestR value is the equilibrium internuclear distance. Notice how the full entropy does not tend to the net entropy, but to the net entropy plus the nonadditive-intra entropy, as predicted alg… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that Shannon and Rényi entropies computed from the electron density and shape function are unreliable descriptors of static electron correlation. Using a Mulliken-like atomic partition to decompose the entropies into additive and non-additive contributions, the authors perform algebraic analysis and numerical calculations at the infinite-separation limit across minimal and larger basis sets and methods (HF vs. full-valence CAS). They report that these density-based measures fail to encode the static correlation present in the wavefunction, that shape-function Rényi entropies violate extensivity, and that HF densities overestimate entropy relative to correlated methods; the conclusion is that robust entropic descriptors require higher-dimensional Hilbert-space objects.

Significance. If the central findings hold after addressing partition dependence, the work would usefully caution against over-reliance on one-electron density entropies for correlation diagnostics and motivate development of wavefunction- or reduced-density-matrix-based alternatives. The algebraic decomposition and infinite-separation asymptotics constitute a clear, falsifiable framework that could be extended to other information measures.

major comments (2)
  1. [Abstract and decomposition section] The headline claim that density-based entropies 'fail to encode' static correlation depends on the Mulliken-like atomic partition and the infinite-separation asymptotic analysis (Abstract and the decomposition procedure). Because this partition is known to be basis-set dependent and can mix delocalization errors with correlation signatures, the non-additivity observed may be an artifact of the chosen decomposition rather than an intrinsic limitation of the electron density. Alternative partitions (Hirshfeld, Bader, or Voronoi) should be tested on the same systems to establish whether the reported failure is robust.
  2. [Numerical results and tables] The numerical evidence that HF overestimates entropy relative to CAS and that insufficiently correlated methods violate extensivity is presented for minimal-basis and larger-basis calculations, but the manuscript does not report error bars, convergence with respect to active-space size, or a quantitative measure of how much static correlation is missed by the entropy values. Without these controls, it is difficult to judge whether the observed discrepancies are decisive or merely reflect the known limitations of HF versus CAS.
minor comments (1)
  1. [Abstract] The abstract states that 'shape-function Shannon entropies and Rényi entropies (for α ≠ 1) violate extensivity'; the precise definition of the shape function and the value of α used in the Rényi calculations should be stated explicitly in the main text for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the major comments below, providing clarifications and indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and decomposition section] The headline claim that density-based entropies 'fail to encode' static correlation depends on the Mulliken-like atomic partition and the infinite-separation asymptotic analysis (Abstract and the decomposition procedure). Because this partition is known to be basis-set dependent and can mix delocalization errors with correlation signatures, the non-additivity observed may be an artifact of the chosen decomposition rather than an intrinsic limitation of the electron density. Alternative partitions (Hirshfeld, Bader, or Voronoi) should be tested on the same systems to establish whether the reported failure is robust.

    Authors: The Mulliken-like partition was selected specifically because it enables an exact algebraic decomposition of the entropies into additive and non-additive parts that is particularly transparent in the infinite-separation limit. This allows us to demonstrate analytically that the non-additive contributions do not reflect the static correlation encoded in the wavefunction. Although Mulliken charges are basis-set dependent, our numerical results show the same qualitative failure across both minimal and larger basis sets. We recognize that other partitions could yield different numerical values; however, the core algebraic argument that density-based entropies cannot capture multi-reference character without higher-order information remains independent of the partition. To address the referee's concern, we will include in the revised manuscript a brief discussion of partition dependence and perform additional calculations using the Hirshfeld partition for the key systems to verify robustness. revision: partial

  2. Referee: [Numerical results and tables] The numerical evidence that HF overestimates entropy relative to CAS and that insufficiently correlated methods violate extensivity is presented for minimal-basis and larger-basis calculations, but the manuscript does not report error bars, convergence with respect to active-space size, or a quantitative measure of how much static correlation is missed by the entropy values. Without these controls, it is difficult to judge whether the observed discrepancies are decisive or merely reflect the known limitations of HF versus CAS.

    Authors: The calculations are deterministic, so error bars are not applicable in the usual sense; the differences between HF and full-valence CAS are systematic and exceed any numerical precision issues. The full-valence CAS represents the complete active space for the valence electrons in the systems considered, providing the appropriate benchmark for static correlation. We will add a quantitative measure by reporting the difference in entropy values normalized to the static correlation energy or by including the configuration weights from the CAS wavefunction to illustrate the extent of the missed correlation. Additionally, we will clarify in the text that the extensivity violation is evident from the non-vanishing non-additive terms at large separations for HF, while they approach zero for CAS. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines a Mulliken-like atomic partition to decompose Shannon and Rényi entropies into additive and non-additive contributions, then derives their infinite-separation asymptotics algebraically and evaluates them numerically for HF versus CAS wavefunctions. These steps rely on established quantum-chemistry methods and partitions without fitting any parameters to the target correlation or extensivity quantities, without self-referential definitions that equate the claimed insufficiency to the input decomposition, and without load-bearing self-citations or smuggled ansatzes. The findings that density-based entropies fail to track static correlation therefore emerge from independent comparisons rather than by construction from the entropy definitions themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the validity of the Mulliken-like partition for separating additive and non-additive entropy contributions and on the assumption that the infinite-separation limit cleanly isolates static correlation without additional approximations.

axioms (2)
  • domain assumption Mulliken-like atomic partition of the molecular density is a valid and unbiased way to define additive versus non-additive entropy contributions.
    Invoked to support the decomposition used throughout the asymptotic analysis.
  • domain assumption The infinite-internuclear-distance limit of the entropies directly reflects the static correlation content of the wavefunction.
    Central to the claim that the entropies fail to encode static correlation.

pith-pipeline@v0.9.0 · 5549 in / 1348 out tokens · 35924 ms · 2026-05-10T05:19:14.166745+00:00 · methodology

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