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arXiv:2508.21578 · The Journal of Chemical Physics

Electronic–nuclear entanglement in Born–Oppenheimer wave functions and beyond

Juan F. P. Mosquera, José Luis Sanz‐Vicario

Published Feb 10, 2026• 0 citations
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Abstract

We analyze the entanglement between electronic and nuclear motions in molecular wave functions widely used by theoretical chemists, namely, (i) Born-Oppenheimer factorization in the adiabatic picture, (ii) the transformation into a diabatic picture, (iii) the use of a Born-Huang expansion, and (iv) the eigenfunction of the full molecular Hamiltonian. Our showcase is based on two one-electron one-dimensional molecular Hamiltonians (H2+ and the Shin-Metiu model). We find that within the Born-Oppenheimer approximation, any molecular state (although factorizable) is always entangled, and its entanglement content may be assessed by the variation of the electronic wave function along the different nuclear geometries, with the nuclear wave function indeed playing the role of a tester. The presence of avoided crossings among the adiabatic potential energy curves brings about dramatic changes in the entanglement content of the wave function: sharp avoided crossings favor a diabatic picture (real crossings between potential energy curves), while in broad avoided crossings, the adiabatic picture prevails. The total eigenfunction of the molecular Shin-Metiu Hamiltonian indicates that nuclear densities accommodate well within the diabatic curves for strong adiabatic couplings but within adiabatic curves for weak ones. Consequently, we find that the electron-nuclei entanglement content is a valid witness to unveil strong or weak nonadiabatic couplings in molecules. In terms of entanglement, we also find that the Born-Huang expansion, based on Born-Oppenheimer adiabatic electronic states, does not provide a correct trend of entanglement compared with that of the total molecular eigenfunction, thus indicating a very slow convergence of this expansion.

DOI
10.1063/5.0311735

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Degree of electron-nuclear entanglement in molecular states

Erik Sj�qvist

The degree of electron–nuclear entanglement in molecular states is analyzed. This entanglement has, generally, two sources: delocalization of the electronic and nuclear wave functions and vibronic coupling. For a diatomic molecular ground-state with a single potential energy minimum, it is demonstrated that the entanglement is a function of the product of the vibrational energy and the Born–Huang potential energy correction evaluated at the minimum. In the case of a double-well potential energy surface, the deviation from maximal entanglement is determined by the overlap of the electronic and nuclear wave functions evaluated at and around the two minima. The adiabatic states of the E⊗ϵ Jahn–Teller model are shown to be maximally entangled and a relation between the degree of entanglement and Ham's reduction factor for this model is derived. Numerical calculations in the E⊗ϵ model demonstrate a nontrivial relation between entanglement and vibronic coupling. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 526–533, 2000

International Journal of Quantum Chemistry•Jan 1, 2000•3 citations
We analyze the entanglement between electronic and nuclear motions in molecular wave functions widely used by theoretical chemists, namely, (i) Born-Oppenheimer factorization in the adiabatic picture, (ii) the transformation into a diabatic picture, (iii) the use of a Born-Huang expansion, and (iv) the eigenfunction of the full molecular Hamiltonian. Our showcase is based on two one-electron one-dimensional molecular Hamiltonians (H2+ and the Shin-Metiu model). We find that within the Born-Oppenheimer approximation, any molecular state (although factorizable) is always entangled, and its entanglement content may be assessed by the variation of the electronic wave function along the different nuclear geometries, with the nuclear wave function indeed playing the role of a tester. The presence of avoided crossings among the adiabatic potential energy curves brings about dramatic changes in the entanglement content of the wave function: sharp avoided crossings favor a diabatic picture (real crossings between potential energy curves), while in broad avoided crossings, the adiabatic picture prevails. The total eigenfunction of the molecular Shin-Metiu Hamiltonian indicates that nuclear densities accommodate well within the diabatic curves for strong adiabatic couplings but within adiabatic curves for weak ones. Consequently, we find that the electron-nuclei entanglement content is a valid witness to unveil strong or weak nonadiabatic couplings in molecules. In terms of entanglement, we also find that the Born-Huang expansion, based on Born-Oppenheimer adiabatic electronic states, does not provide a correct trend of entanglement compared with that of the total molecular eigenfunction, thus indicating a very slow convergence of this expansion.
ABSTRACT: In the present work, we have computed the entanglement between the electronic and nuclear motions in two molecular model systems: the one-dimensional hydrogen molecular ion (H2+) and the Shin-Metiu model, considering the molecules as a bipartite systems: electron and nuclear motion. For that purpose, we have computed the Born-Oppenheimer and non-Born-Oppenheimer (Born-Huang) wave function in terms of the Fourier Grid Hamiltonian basis that expands both the electronic and nuclear wave functions. Also, according to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, there is a much shorter but equivalent expansion in terms of the Schmidt bases for the electronic and nuclear sub-spaces. In these models of distinguishable coupled particles we have shown that the entanglements contents do not increase monotonically with the excitation energy. In the hydrogen molecular ion and in the ShinMetiu model, the entanglements contents for each Born-Oppenheimer electro-nuclear state is quantified through the von-Neumann and linear entropies and we have shown that entanglement serves as a witness of distinguishability of nuclear states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.
We compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.
The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. Although nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.
The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. While nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.
Over the last few decades, Born-Oppenheimer (BO) and Born-Oppenheimer-Huang (BOH) treatments for electron-nuclear equation of motion have been considered as the cornerstone for qualitative as well as quantitative exploration and understanding of chemical and physical processes. The BO approximation due to the large mass difference between electrons and nuclei as well as the energy separation among the electronic states could able to predict some of the processes (chemical and physical) satisfactorily at lower energy regime. Such approximation (BO) quite often fails for those cases with the involvement of excited electronic states, in particular, due to the coupling among the states. On the other hand, the BO treatment provides two important quantities, namely, the adiabatic potential energy surfaces (PESs) and the nonadiabatic coupling terms (NACTs) determining the forces, which control the motion of the nuclei in a molecule. The NACTs play a major role in photoexcitation spectra, charge transfer chemical reactions, ferroelectric property and other processes, where significant anomalies are observed between single and multiple PES(s) results. Since the NACTs often exhibit singularity at the close vicinity of degenerate states forming conical intersections (CIs), such coupling terms could break the symmetry of the molecular processes changing the observables qualitatively.
The degree of electron–nuclear entanglement in molecular states is analyzed. This entanglement has, generally, two sources: delocalization of the electronic and nuclear wave functions and vibronic coupling. For a diatomic molecular ground-state with a single potential energy minimum, it is demonstrated that the entanglement is a function of the product of the vibrational energy and the Born–Huang potential energy correction evaluated at the minimum. In the case of a double-well potential energy surface, the deviation from maximal entanglement is determined by the overlap of the electronic and nuclear wave functions evaluated at and around the two minima. The adiabatic states of the E⊗ϵ Jahn–Teller model are shown to be maximally entangled and a relation between the degree of entanglement and Ham's reduction factor for this model is derived. Numerical calculations in the E⊗ϵ model demonstrate a nontrivial relation between entanglement and vibronic coupling. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 526–533, 2000