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The Journal of Physical Chemistry Letters

Orbital Entanglement Analysis of Exchange-Coupled Systems

Christopher Johannes Stein, Dimitrios A. Pantazis, Vera Krewald

Published Oct 15, 2019• 23 citations
Publisher

Abstract

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.

DOI
10.1021/acs.jpclett.9b02417

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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 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.
The applicability of ab initio multireference wavefunction-based methods to the study of magnetic complexes has been restricted by the quickly rising active-space requirements of oligonuclear systems and dinuclear complexes with S > 1 spin centers. Ab initio density matrix renormalization group (DMRG) methods built upon an efficient parameterization of the correlation network enable the use of much larger active spaces, and therefore may offer a way forward. Here, we apply DMRG-CASSCF to the dinuclear complexes [Fe2OCl6](2-) and [Cr2O(NH3)10](4+). After developing the methodology through systematic basis set and DMRG M testing, we explore the effects of extended active spaces that are beyond the limit of conventional methods. We find that DMRG-CASSCF with active spaces including the metal d orbitals, occupied bridging-ligand orbitals, and their virtual double shells already capture a major portion of the dynamic correlation effects, accurately reproducing the experimental magnetic coupling constant (J) of [Fe2OCl6](2-) with (16e,26o), and considerably improving the smaller active space results for [Cr2O(NH3)10](4+) with (12e,32o). For comparison, we perform conventional MRCI+Q calculations and find the J values to be consistent with those from DMRG-CASSCF. In contrast to previous studies, the higher spin states of the two systems show similar deviations from the Heisenberg spectrum, regardless of the computational method.
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.
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.
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.
Model Hamiltonians (e.g., the Hubbard or Heisenberg models) provide a simple yet physically meaningful description of complex electronic phenomena through a small number of parameters, such as electron hopping integrals or magnetic exchange couplings. Their construction typically exploits the local nature of electron correlation effects and therefore relies on localized orbitals. Although many localization schemes are available, the resulting localized orbitals depend not only on the chosen localization functional but also on the orbital space to which the localization procedure is applied. These choices are often treated as technical details and are rarely discussed explicitly. Here, we demonstrate that different, chemically reasonable choices of localized orbitals can lead to substantially different model Hamiltonian parameters, even when they produce nearly identical electronic energies. Using Density Matrix Downfolding, we systematically investigate how the localization functional and, more importantly, the choice of model orbital space affect effective Hamiltonians derived from ab initio calculations for representative πππ-conjugated systems and transition metal complexes.