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Journal of Modern Optics

Entanglement and electron correlation in quantum chemistry calculations

Zhen Huang, H. Wang, Sabre Kais

Published Oct 25, 2006• 56 citations
Publisher

Abstract

Electron–electron correlation in quantum chemistry calculations can be analysed in terms of entanglement between electrons. Two exactly solvable models: two fixed spin-1/2 particles and two-electron two-site Hubbard model are used to define and discuss the entanglement as a function of the system parameters. Ab initio configuration interaction calculation for entanglement is presented for the H2 molecule. Qualitatively, entanglement and electron–electron correlation have similar behaviour. Thus, entanglement might be used as an alternative measure of electron correlation in quantum chemistry calculations.

DOI
10.1080/09500340600955674

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Electron–electron correlation in quantum chemistry calculations can be analysed in terms of entanglement between electrons. Two exactly solvable models: two fixed spin-1/2 particles and two-electron two-site Hubbard model are used to define and discuss the entanglement as a function of the system parameters. Ab initio configuration interaction calculation for entanglement is presented for the H2 molecule. Qualitatively, entanglement and electron–electron correlation have similar behaviour. Thus, entanglement might be used as an alternative measure of electron correlation in quantum chemistry calculations.
Abstract We study the relation between quantum entanglement and electron correlation in quantum chemistry calculations. We prove that the Hartree–Fock (HF) wave function does not violate Bell's inequality, and thus is not entangled, whereas the configuration interaction (CI) wave function is entangled since it violates Bell's inequality. Entanglement is related to electron correlation and might be used as an alternative measure of the electron correlation in quantum chemistry calculations. As an example we show the calculations of entanglement for the H2 molecule and how it correlates with the traditional electron correlation, which is the difference between the exact and the HF energies.
We study the relation between quantum entanglement and electron correlation in quantum chemistry calculations. We prove that the Hartree-Fock (HF) wave function does not violate Bell's inequality, thus is not entangled while the configuration interaction (CI) wave function is entangled since it violates Bell's inequality. Entanglement is related to electron correlation and might be used as an alternative measure of the electron correlation in quantum chemistry calculations. As an example we show the calculations of entanglement for the H2_22​ molecule and how it is related to electron correlation of the system, which is the difference between the exact and the HF energies.
We use the concept of quantum entanglement to give a physical meaning to the electron correlation energy in systems of interacting electrons. The electron correlation is not directly observable, being defined as the difference between the exact ground state energy of the many-electron Schr ̈odinger equation and the Hartree–Fock energy. Using the configuration interaction method for the hydrogen molecule, we calculate the correlation energy and compare it with the entanglement as a function of the nucleus– nucleus separation. In the same spirit, we analyze a dimer of ethylene, which represents the simplest organic conjugate system, changing the relative orientation and distance of the molecules to obtain the configuration corresponding to maximum entanglement.
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.
Recent proposals of measuring bipartite Renyi entropy experimentally involve techniques that hold exactly for non-interacting quantum particles. Here we consider the difference between such measurements and the actual Renyi entropy for ground state fermion ab initio molecular systems. To calculate various entanglement measures we extended several different techniques for use with variational Monte Carlo, Hartree-Fock and quantum chemistry methods. Our results show that in systems with strong electron correlations the Renyi entropy may not be accurately determined with the proposed measurements. In addition, we find significant physical insight can be gained by calculating entanglement properties in molecular systems. We see that the Renyi entropy and the entanglement Hamiltonian encode information about the character of the covalent bonds in a molecular system and that such information may lead to better descriptions of bonded systems that have been traditionally hard to describe with standard techniques. In particular our results for the C2_{2}2​ dimer suggests all eight valence electrons play a significant role in covalent bonding.
By going beyond Hubbard Hamiltonian we reflected correlation effects accurately in the wavefunctions of H2. Using ab initio e-e interaction parameters resulted maximally entangled ground and third excited states. We assigned this maximally entangled character to the nonmagnetic (S=0) property of these states and also the minimally entangled character of the first excited states to its magnetic property. By switching on a magnetic field an entangled state with Sz = 0 can be extracted from a minimally entangled degenerate magnetic state. We suggest that presence of a moderate correlated system and a non-magnetic (Sz = 0) electronic state can be two criteria for finding maximally entangled electronic states in a realistic molecular system.