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arXiv:2605.25793

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

Jiřı́ Pittner

Published May 25, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2605.25793

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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 (MsM_sMs​) 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 MsM_sMs​. 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 MsM_sMs​ 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.
Orbital entropies, pair entropies and mutual information have become popular tools for the analysis of strongly correlated wave functions. They can quantitatively measure how strongly an orbital (e.g. from the DMRG active space) participates in the strong correlation and reveals 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 (Ms) 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 Ms. By comparing 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. We illustrate the approach on a model consisting of a non-interacting dimer of triplet diradicals and on a more realistic example of iron-sulphur bound complexes with one and two iron atoms.
High Resolution Image Download MS PowerPoint Slide Quantifying correlation and complexity in quantum many-body states is central to advancing theoretical and computational chemistry, physics, and quantum information science. This work introduces a novel framework, mutual correlation, based on the Frobenius norm squared of the two-body reduced density matrix cumulant. Through systematic partitioning of the cumulant norm, mutual correlation quantifies nonadditive correlations among interacting subsystems. To assess the ability of mutual correlation to identify orbital interactions, we performed benchmark studies on model systems, including H 10, N 2, and p -benzyne, and performed a formal and numerical comparison with orbital mutual information. Maximally correlated orbitals, obtained by maximizing a nonlinear cost function of the mutual correlation, are also considered to identify a basis-independent partitioning of correlation. This study suggests that mutual correlation is a broadly applicable metric, useful in active space selection and the interpretation of electronic states.
A new tool for the interpretation of multiconfigurational wave functions representing the spin states of exchange-coupled transition metal complexes is introduced. Based on orbital entanglement measures, herein derived from multiconfigurational density matrix renormalization group calculations, the complexity of the wave function is reduced, thus facilitating a connection with established concepts for the interpretation of magnetically coupled systems. We show that the entanglement of localized orbitals with a small basis set is a good representation of the magnetic coupling topology and that it is sensitive to chemical changes in homologous complexes. Furthermore, we introduce a measure for the magnetic relevance of orbitals in the active subspace and a concept for the quantitative comparison of different chemical species. The approach presented here will be easily applicable to higher nuclearity clusters, providing a direct insight into all states of the Heisenberg spin ladder for systems previously accessible only by single-configurational methods.
The use of the mutual information (MI) as a measure of the entanglement in quantum systems has gained a consensus in recent years, even if there is an ongoing effort to distinguish the classical and quantum contributions contained therein. This quantity has been first introduced in condensed matter physics, in particular, in studies based on the density matrix renormalization group method. This method has been successfully adapted to quantum chemistry problems, opening the way to compute MI also in molecular systems. A key aspect of this quantity is its dependence on the one-electron (orbital) basis set, even for wave functions that are invariant under unitary transformation of the orbitals. In this work, we investigate the role of the orbital basis set (delocalized or localized, following different strategies) for wave functions expressed as linear combinations of Slater determinants and we give the analytic expression for the MI for a few special cases. This study aims to improve the knowledge of the relationship between the characteristics of the chemical bond (considering a few paradigmatic molecules, H 2, F 2, N 2, and short linear polyenes) and the properties of interest in the field of quantum information theory.
Pair density Shannon information measures and their NNN-dependent behaviour in diatomic molecular systems
Pair density Shannon entropies and mutual information in position and in momentum space are calculated for two series of homonuclear and heteronuclear diatomic molecules using Hartree-Fock wave functions. The pair density entropy sum increases with the quality of the basis set. Mutual information, a measure of statistical correlation, is seen to be smaller in momentum space as compared to position space. The interpretation is that the momentum pair density is closer than the position pair density to a Hartree-like reference. The NNN-dependent behaviours of the entropy and mutual information sums are examined by fitting the data to three different model behaviours. Results show that these models are capable of representing the molecular data in differing degrees. The parameters obtained from a particular model are relatively constant across different chemical series. This adds evidence to the argument of a universal NNN-dependent behaviour of the entropy sums.
Wave functions constructed from electron-pair states can accurately model strong electron correlation effects and are promising approaches especially for larger many-body systems. In this article, we analyze the nature and the type of electron correlation effects that can be captured by wave functions restricted to electron-pair states. We focus on the pair-coupled-cluster doubles (pCCD) ansatz also called the antisymmetric product of the 1-reference orbital geminal (AP1roG) method, combined with an orbital optimization protocol presented in Boguslawski et al. [Phys. Rev. B 89, 201106(R) (2014)], whose performance is assessed against electronic structures obtained form density-matrix renormalization-group reference data. Our numerical analysis covers model systems for strong correlation: the one-dimensional Hubbard model with a periodic boundary condition as well as metallic and molecular hydrogen rings. Specifically, the accuracy of pCCD/AP1roG is benchmarked using the single-orbital entropy, the orbital-pair mutual information, as well as the eigenvalue spectrum of the one-orbital and two-orbital reduced density matrices. Our study indicates that contributions from singly occupied states become important in the strong correlation regime which highlights the limitations of the pCCD/AP1roG method. Furthermore, we examine the effect of orbital rotations within the pCCD/AP1roG model on correlations between orbital pairs.