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arXiv:2505.02930 · The Journal of Chemical Physics

Orbital entanglement and the double d-shell effect in binary transition metal molecules

Julianne S. Lampert, Timothy J. Krogmeier, Anthony W. Schlimgen, Kade Head-Marsden

Published Nov 1, 2024• 4 citations
arXivHTMLPDF

Abstract

Accurate modeling of transition metal-containing compounds is of great interest due to their wide-ranging and significant applications. These systems present several challenges from an electronic structure perspective, including significant multi-reference characters and many chemically relevant orbitals. A further complication arises from the so-called double d-shell effect, which is known to cause a myriad of issues in the treatment of first-row transition metals with both single- and multi-reference methods. While this effect has been well documented for several decades, a comprehensive understanding of its consequences and underlying causes is still evolving. Here, we characterize the second d-shell effect by analyzing the information entropy of correlated wavefunctions in a periodic series of 3d and 4d transition metal molecular hydrides and oxides. These quantum information techniques provide unique insight into the nuanced electronic structure of these species and are powerful tools for the study of weak and strong correlations in the transition metal d manifold.

DOI
10.1063/5.0232316

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Accurate modeling of transition metal-containing compounds is of great interest due to their wide-ranging and significant applications. These systems present several challenges from an electronic structure perspective, including significant multi-reference characters and many chemically relevant orbitals. A further complication arises from the so-called double d-shell effect, which is known to cause a myriad of issues in the treatment of first-row transition metals with both single- and multi-reference methods. While this effect has been well documented for several decades, a comprehensive understanding of its consequences and underlying causes is still evolving. Here, we characterize the second d-shell effect by analyzing the information entropy of correlated wavefunctions in a periodic series of 3d and 4d transition metal molecular hydrides and oxides. These quantum information techniques provide unique insight into the nuanced electronic structure of these species and are powerful tools for the study of weak and strong correlations in the transition metal d manifold.
Electron correlation effects are essential for an accurate ab initio description of molecules. A quantitative a priori knowledge of the single- or multi-reference nature of electronic structures as well as of the dominant contributions to the correlation energy can facilitate the decision regarding the optimum quantum chemical method of choice. We propose concepts from quantum information theory as orbital entanglement measures that allow us to evaluate the single- and multi-reference character of any molecular structure in a given orbital basis set. By studying these measures we can detect possible artifacts of small active spaces.
We have studied transition metal clusters from a quantum information theory perspective using the density-matrix renormalization group (DMRG) method. We demonstrate the competition between entanglement and interaction localization and discuss the application of the configuration interaction-based dynamically extended active space procedure, which significantly reduces the effective system size and accelerates the speed of convergence for complicated molecular electronic structures. Our results indicate the importance of taking entanglement among molecular orbitals into account in order to devise an optimal DMRG orbital ordering and carry out efficient calculations on transition metal clusters. Apart from these algorithmic observations, which lead to a recipe for black-box DMRG calculations, our work provides physical understanding of electron correlation in molecular and cluster structures in terms of entropy measures of relevance also to recent work on tensor-network representations of electronic states. We also identify those molecular orbitals which are highly entangled and discuss the consequences for chemical bonding and for the structural transition from an dioxygen binding copper cluster to an bis-oxygen-bridged system with broken O--O bond.
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Electron correlation effects are essential for an accurate ab initio description of molecules. A quantitative a priori knowledge of the single- or multireference nature of electronic structures as well as of the dominant contributions to the correlation energy can facilitate the decision regarding the optimum quantum chemical method of choice. We propose concepts from quantum information theory as orbital entanglement measures that allow us to evaluate the single- and multireference character of any molecular structure in a given orbital basis set. By studying these measures we can detect possible artifacts of small active spaces.
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Abstract Quantum entanglement offers a unique perspective into the underlying structure of strongly-correlated systems such as atomic nuclei. In this paper, we use quantum information tools to analyze the structure of light and medium-mass berillyum, oxygen, neon and calcium isotopes within the nuclear shell model. We use different entanglement metrics, including single-orbital entanglement, mutual information, and von Neumann entropies for different equipartitions of the shell-model valence space and identify mode-entanglement patterns related to the energy, angular momentum and isospin of the nuclear single-particle orbitals. We observe that the single-orbital entanglement is directly related to the number of valence nucleons and the energy structure of the shell, while the mutual information highlights signatures of proton–proton and neutron–neutron pairing, as well as nuclear deformation. Proton and neutron orbitals are weakly entangled by all measures, and in fact have the lowest von Neumann entropies among all possible equipartitions of the valence space. In contrast, orbitals with opposite angular momentum projection have relatively large entropies, especially in spherical nuclei. This analysis provides a guide for designing more efficient quantum algorithms for the noisy intermediate-scale quantum era.