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arxiv: 2605.25793 · v1 · pith:GSK6FS57new · submitted 2026-05-25 · ⚛️ physics.chem-ph

Mutual information and mutual correlation: their spin-free formulations and comparison

Jiri Pittner This is my paper

Pith reviewed 2026-06-29 19:31 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords orbital entropypair entropymutual informationmutual correlationspin-free formulationtwo-body cumulantelectron correlationiron-sulfur complexes
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The pith

Spin-free versions of orbital entropy and mutual information are invariant to spin projection Ms and separate spin-coupling correlation from multiconfigurational effects.

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

The paper introduces modified spin-free orbital entropy, pair entropy, and mutual information that remain unchanged regardless of the Ms component of a spin multiplet. These quantities simplify correlation pattern analysis in strongly correlated wave functions and, when compared to their original spin-including versions, distinguish static correlation arising from spin couplings from genuine strong correlation due to multiconfigurational character. A spin-free analogue of mutual correlation computed from the two-body cumulant is also defined as an alternative measure easier to obtain for some methods. The entropy-based and cumulant-based approaches are compared in both spin-free and original forms. The methods are illustrated through applications to iron-sulfur bound complexes with one and two iron atoms.

Core claim

The central claim is that modified spin-free orbital entropy, pair entropy, and mutual information provide an Ms-invariant framework for analyzing electron correlation patterns; comparison with the standard versions separates correlation due to spin couplings from multiconfigurational strong correlation, while a parallel spin-free mutual correlation derived from the two-body cumulant enables direct comparison of entropy-based and cumulant-based measures on the same systems.

What carries the argument

Modified spin-free orbital entropy, pair entropy, and mutual information (together with their cumulant-based counterpart), which achieve Ms invariance by reformulation that removes dependence on the spin projection component.

If this is right

  • Correlation patterns become simpler to interpret for large active spaces because the measures no longer vary with Ms.
  • Direct comparison of spin-free and original mutual information isolates the contribution of spin couplings to the observed correlation.
  • The spin-free cumulant-based mutual correlation supplies an alternative that is computationally lighter for certain electronic structure methods.
  • Both entropy-based and cumulant-based measures can be evaluated side-by-side in their spin-free and original forms on the same wave function.
  • The framework applies directly to the analysis of iron-sulfur complexes containing one or two metal centers.

Where Pith is reading between the lines

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

  • The Ms-invariant measures could be used to compare correlation across different spin states of the same molecule without additional adjustments.
  • Routine use of these quantities might guide the choice of active space by highlighting orbitals whose correlation is independent of spin projection.
  • The distinction between the two types of correlation may help decide when a single-reference method suffices versus when a multireference treatment is required.

Load-bearing premise

Comparing the spin-free and original quantities on iron-sulfur complexes reliably separates spin-coupling correlation from multiconfigurational correlation without the separation depending on the specific active space or method chosen.

What would settle it

Observing that the patterns distinguishing spin-coupling from multiconfigurational correlation change substantially when the same iron-sulfur complex is recomputed with a different active space or different method would falsify the reliability of the distinction.

Figures

Figures reproduced from arXiv: 2605.25793 by Jiri Pittner.

Figure 1
Figure 1. Figure 1: Structure of the [Fe(SCH3)4]− and [Fe2S2(SCH3)4] 2− complexes employed the Orca program [67] and the Def2-TZVP basis set to perform a configuration-averaged ROHF SCF calculations and integral dump. The MOLMPS code[68] was used for the DMRG calculations of orbital density matrices, while the BLOCK2 code[69] was employed for the reduced density matrices (due to better numerical stability for these quantities… view at source ↗
Figure 2
Figure 2. Figure 2: Correlation analysis for the ground state (S = 5/2) of the [Fe(SCH3)4]− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantiti… view at source ↗
Figure 3
Figure 3. Figure 3: Correlation analysis for the state (S = 3/2) of the [Fe(SCH3)4]− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantities are … view at source ↗
Figure 4
Figure 4. Figure 4: Correlation analysis for the state (S = 1/2) of the [Fe(SCH3)4]− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantities are … view at source ↗
Figure 5
Figure 5. Figure 5: Scatter plots correlating orbital entropy, Von Neumann orbital from from 1RDM, and orbital sum of mutual correlation. The quantities on the y axis were renormalized to match the norm of the orbital entropy vector (over all orbitals). Colors code the state: S = 5/2 red, S = 3/2 orange, S = 1/2 gray. Configuration-averaged ROHF orbitals have been employed. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Scatter plots correlating mutual information and mutual correlation. The quantities on the y axis were renormalized to match the norm of the mutual information vector (over all pairs orbitals). Colors code the state: S = 5/2 red, S = 3/2 orange, S = 1/2 gray. Configuration-averaged ROHF orbitals have been employed [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Correlation analysis for the S = 0 (ground) state of the [Fe2S2(SCH3)4] 2− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quant… view at source ↗
Figure 8
Figure 8. Figure 8: Correlation analysis for the S = 1 state of the [Fe2S2(SCH3)4] 2− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantities are… view at source ↗
Figure 9
Figure 9. Figure 9: Correlation analysis for the S = 2 state of the [Fe2S2(SCH3)4] 2− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantities are… view at source ↗
Figure 10
Figure 10. Figure 10: Correlation analysis for the S = 3 state of the [Fe2S2(SCH3)4] 2− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantities ar… view at source ↗
Figure 11
Figure 11. Figure 11: Correlation analysis for the S = 4 state of the [Fe2S2(SCH3)4] 2− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantities ar… view at source ↗
Figure 12
Figure 12. Figure 12: Correlation analysis for the S = 5 state of the [Fe2S2(SCH3)4] 2− complex, based on mutual information (left) and mutual correlation (right). The bar graphs shows orbital entropies (left) or orbital sum of mutual information (right), while the color-coded size of the mutual information (left) and mutual correlation (right) matrix elements is displayed below. The weighted graphs combining the quantities ar… view at source ↗
Figure 13
Figure 13. Figure 13: Scatter plots correlating orbital entropy, Von Neumann orbital from from 1RDM, and orbital sum of mutual correlation. The quantities on the y axis were renormalized to match the norm of the orbital entropy vector (over all orbitals). Colors code the state: S = 0 red, S = 1 blue, S = 2 violet, S = 3 yellow, S = 4 pink, S = 5 gray [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Scatter plots correlating mutual information and mutual correlation. The quantities on the y axis were renormalized to match the norm of the mutual information vector (over all pairs orbitals). Colors code the state: S = 0 red, S = 1 blue, S = 2 violet, S = 3 yellow, S = 4 pink, S = 5 gray. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
read the original abstract

Orbital entropies, pair entropies, and mutual information have become popular tools for analysis of strongly correlated wave functions. They can quantitatively measure how strongly an orbital participates in the electron correlation and reveal the correlation pattern between different orbitals. However, this pattern can become rather complicated and sometimes difficult to interpret for large active spaces and is not invariant with respect to the spin projection ($M_s$) component of the spin multiplet state. We introduce a modified spin-free orbital entropy, pair entropy, and mutual information, which simplify the correlation analysis and are invariant with respect to $M_s$. By comparison of these quantities with their "original" spin-including counterparts one can distinguish static correlation due to spin couplings from the "genuine" strong correlation due to a multiconfigurational character of the wave function. Recently, Evangelista introduced mutual correlation computed from the two-body cumulant as an alternative correlation measure, which is easier to compute for some methods. We present here a spin-free analogue of this quantity, which is $M_s$ invariant, and perform a comparison of the entropy-based and cumulant-based correlation measures in both spin-free and original variants. We illustrate the approaches on iron-sulfur bound complexes with one and two iron atoms.

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 / 2 minor

Summary. The manuscript introduces Ms-invariant spin-free formulations of orbital entropy, pair entropy, and mutual information, along with a spin-free analogue of Evangelista's cumulant-based mutual correlation. It claims that differences between these quantities and their original spin-dependent counterparts distinguish static correlation arising from spin couplings from genuine strong correlation due to multiconfigurational character, with numerical illustrations on iron-sulfur complexes containing one or two Fe atoms.

Significance. If the interpretive distinction holds, the work supplies a practical, Ms-invariant tool for simplifying correlation analysis in open-shell systems with large active spaces, which is relevant to transition-metal and bioinorganic chemistry. The explicit comparison between entropy-based and cumulant-based measures, together with the spin-free reformulations, adds a useful diagnostic layer beyond existing entropy tools.

major comments (2)
  1. [§5] §5 (applications to [Fe(SCH3)4]− and [Fe2S2] clusters): the numerical differences between spin-free and original mutual information are presented for fixed active spaces and CASSCF/DMRG wave functions, but no systematic variation of active-space size or orbital selection is reported; this leaves open whether the claimed separation of spin-coupling versus multiconfigurational effects is robust or depends on the specific active-space choice.
  2. [§3] §3 (spin-free definitions): while the Ms-invariant expressions for orbital/pair entropy and mutual information are given, the manuscript does not derive or prove that the numerical difference between spin-free and original quantities isolates spin-coupling effects independently of the underlying 1- and 2-RDMs; a short algebraic argument or counter-example test would strengthen the central interpretive claim.
minor comments (2)
  1. [Table 1, Figure 3] Table 1 and Figure 3: the reported mutual-information values would benefit from explicit error bars or convergence checks with respect to DMRG bond dimension to confirm numerical stability.
  2. [§2–3] Notation: the symbol for the spin-free pair entropy is introduced without a clear cross-reference to its spin-dependent counterpart; adding an explicit side-by-side definition table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and positive evaluation of the significance of our work. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [§5] §5 (applications to [Fe(SCH3)4]− and [Fe2S2] clusters): the numerical differences between spin-free and original mutual information are presented for fixed active spaces and CASSCF/DMRG wave functions, but no systematic variation of active-space size or orbital selection is reported; this leaves open whether the claimed separation of spin-coupling versus multiconfigurational effects is robust or depends on the specific active-space choice.

    Authors: The examples in §5 are selected as representative cases from bioinorganic chemistry to illustrate the distinction in practice. We agree that the absence of systematic active-space variation leaves the robustness open to question. In the revised manuscript we will add a paragraph discussing the rationale for the chosen active spaces, report results for one alternative orbital selection per complex where computationally feasible, and explicitly note the limitations of the current numerical evidence. revision: partial

  2. Referee: [§3] §3 (spin-free definitions): while the Ms-invariant expressions for orbital/pair entropy and mutual information are given, the manuscript does not derive or prove that the numerical difference between spin-free and original quantities isolates spin-coupling effects independently of the underlying 1- and 2-RDMs; a short algebraic argument or counter-example test would strengthen the central interpretive claim.

    Authors: We accept the referee’s point that the interpretive claim requires explicit justification. The spin-free quantities are obtained from spin-averaged RDMs, which by construction remove Ms-dependent spin-coupling contributions. We will insert a short algebraic derivation in the revised §3 showing that the difference vanishes identically for any closed-shell (spin-singlet) state and is nonzero precisely when the wave function belongs to a spin multiplet with static spin correlation, independent of the concrete numerical values of the 1- and 2-RDMs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; spin-free reformulations are direct modifications of existing quantities

full rationale

The paper derives spin-free orbital entropy, pair entropy, mutual information, and mutual correlation by explicit algebraic modifications of standard definitions (e.g., averaging over Ms components or removing spin-dependent terms) to enforce Ms invariance. These steps are self-contained mathematical re-expressions with no fitted parameters renamed as predictions, no load-bearing self-citations, and no uniqueness theorems imported from prior author work. The comparison to original quantities and illustration on iron-sulfur complexes is presented as an empirical demonstration rather than a derivation that reduces to its own inputs by construction. The work remains independent of any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the contribution consists of reformulations of standard information-theoretic quantities already used in quantum chemistry.

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Reference graph

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