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arXiv:2501.15699 · Nature Communications

Chemical bonding concepts emerge naturally from maximally entangled atomic orbitals

Lexin Ding, Eduard Matito, Christian Schilling

Published May 27, 2026• 2 citations
arXivHTMLPDF

Abstract

Chemical bonding is a nonlocal phenomenon that binds atoms into molecules. Its ubiquitous presence in chemistry, however, stands in stark contrast to its ambiguous definition and the lack of a universal perspective for its understanding. In this work, we rationalize and characterize chemical bonding through the lens of an equally nonlocal concept from quantum information, the orbital entanglement. We introduce the maximally entangled atomic orbitals (MEAOs) whose entanglement pattern is shown to recover both Lewis (two-center) and beyond-Lewis (multicenter) structures, with multipartite entanglement serving as a comprehensive index of bond strength. Our unifying framework for bonding analyses is effective not only for equilibrium geometries but also for transition states in chemical reactions and complex phenomena such as aromaticity. It also has the potential to elevate the Hilbert space atomic partitioning to match the prevalent real-space partitioning in the theory of atoms in molecules. Accordingly, our work provides a new framework for understanding fuzzy chemical concepts using rigorous, quantitative descriptors from quantum information.

DOI
10.1038/s41467-026-73527-w

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Chemical bonding is a nonlocal phenomenon that binds atoms into molecules. Its ubiquitous presence in chemistry, however, stands in stark contrast to its ambiguous definition and the lack of a universal perspective for its understanding. In this work, we rationalize and characterize chemical bonding through the lens of an equally nonlocal concept from quantum information, the orbital entanglement. We introduce the maximally entangled atomic orbitals (MEAOs) whose entanglement pattern is shown to recover both Lewis (two-center) and beyond-Lewis (multicenter) structures, with multipartite entanglement serving as a comprehensive index of bond strength. Our unifying framework for bonding analyses is effective not only for equilibrium geometries but also for transition states in chemical reactions and complex phenomena such as aromaticity. It also has the potential to elevate the Hilbert space atomic partitioning to match the prevalent real-space partitioning in the theory of atoms in molecules. Accordingly, our work provides a new framework for understanding fuzzy chemical concepts using rigorous, quantitative descriptors from quantum information.
Abstract The second quantum revolution is all about exploiting the quantum nature of atoms and molecules to execute quantum information processing tasks. To boost this growing endeavor and by anticipating the key role of quantum chemistry therein, our work establishes a framework for systematically exploring, quantifying and dissecting correlation effects in molecules. By utilizing the geometric picture of quantum states we compare—on a unified basis and in an operationally meaningful way—total, quantum and classical correlation and entanglement in molecular ground states. To unlock and maximize the quantum informational resourcefulness of molecules an orbital optimization scheme is developed, leading to a paradigm-shifting insight: a single covalent bond equates to the entanglement 2 ln ( 2 ) . This novel and more versatile perspective on electronic structure suggests a generalization of valence bond theory, overcoming deficiencies of modern chemical bonding theories.
A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements in numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement seen previously. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.
The chemical bond is an important local concept to understand chemical compounds and processes. Unfortunately, like most local concepts, the chemical bond and the bond order do not correspond to any physical observable and thus cannot be determined as an expectation value of a quantum chemical operator. We recently demonstrated [Boguslawski et al., J. Chem. Theory Comput., 2013, 9, 2959-2973] that one- and two-orbital-based entanglement measures can be applied to interpret electronic wave functions in terms of orbital correlation. Orbital entanglement emerged as a powerful tool to provide a qualitative understanding of bond-forming and bond-breaking processes, and allowed for an estimation of bond orders of simple diatomic molecules beyond the classical bonding models. In this article we demonstrate that the orbital entanglement analysis can be extended to polyatomic molecules to understand chemical bonding.
Quantum information theory provides a powerful toolbox of descriptors that characterize many-electron systems based on quantum information patterns between open quantum systems. Despite the wealth of insights gained in the condensed matter community, the use of these descriptors to study interactions between atoms in a molecule remains limited. In this study, we develop a quantum information framework for molecules that characterizes the quantum information patterns between quantum atoms as defined in the Quantum Theory of Atoms in Molecules. We show that quantum information analyses capture key properties of quantum atoms and how they interact with their molecular environment. Additionally, we show that the presence of bond critical points can remain invariant despite large changes in the quantum information patterns between the quantum atoms. Our findings indicate that quantum information theory can shed a new light on molecular electronic structure.
The basic concepts of orbital entanglement and its application to chemistry are briefly reviewed. The calculation of orbital entanglement measures from correlated wavefunctions is discussed in terms of reduced n ‐particle density matrices. Possible simplifications in their evaluation are highlighted in case of seniority‐zero wavefunctions. Specifically, orbital entanglement allows us to dissect electron correlation effects in its strong and weak contributions, to determine bond orders, to assess the quality and stability of active space calculations, to monitor chemical reactions, and to identify points along the reaction coordinate where electronic wavefunctions change drastically. Thus, orbital entanglement represents a useful and intuitive tool to interpret complex electronic wavefunctions and to facilitate a qualitative understanding of electronic structure and how it changes in chemical processes. © 2014 Wiley Periodicals, Inc.
Schilling, Ding, and Matito (Nature Communications, 2026) demonstrate that classicalchemical bonding patterns emerge strictly from Maximally Entangled Atomic Orbitals (MEAOs).We prove that their information-theoretic procedure is mathematically isomorphic to orthogonal projection onto a lower-dimensional persistent sparse invariant manifold. This establishes the Representation Boundary Principle across both quantum chemistry and hydrodynamics: continuous unitary representations must be replaced by discrete topological projections to recover physically interpretable persistent structures Ψ. The Deterministic Spectral Manifold (DSM-861) provides one arithmetically rigid realization of such a projection space, unifying the mathematical treatment of macroscopic fluid singularities and subatomic chemical bonds