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Projected site-occupation embedding theory: a rigorous embedding combining wavefunction theory and density functional theory

Bruno Senjean

Published Feb 15, 2019• 0 citations
OpenAlex

Abstract

Site-occupation embedding theory (SOET) [B. Senjean et al., Theo. Chem. Acc. (2018) 137:169] is an in-principle exact embedding method combining wavefunction theory and density functional theory that yields promising results when applied to the one-dimensional Hubbard model. However the embedded problem in SOET remains the size of the full system, which is problematic as the computational cost increases exponentially with system size. In this work, this issue is circumvented by employing the Schmidt decomposition, thus leading to a significant reduction of the computational cost while conserving the same accuracy. We show that this projected implementation of SOET is competitive with other embedding techniques such as density matrix embedding theory [G. Knizia and G. K-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)] and density embedding theory [I. Bulik et al. Phys. Rev. B 89, 035140 (2014)]. As an important result, the density-driven Mott-Hubbard transition has been predicted with a single impurity site, which is a first for this type of embedding methods.

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Projected site-occupation embedding theory

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Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.

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Site-occupation embedding theory (SOET) [B. Senjean et al., Theo. Chem. Acc. (2018) 137:169] is an in-principle exact embedding method combining wavefunction theory and density functional theory that yields promising results when applied to the one-dimensional Hubbard model. However the embedded problem in SOET remains the size of the full system, which is problematic as the computational cost increases exponentially with system size. In this work, this issue is circumvented by employing the Schmidt decomposition, thus leading to a significant reduction of the computational cost while conserving the same accuracy. We show that this projected implementation of SOET is competitive with other embedding techniques such as density matrix embedding theory [G. Knizia and G. K-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)] and density embedding theory [I. Bulik et al. Phys. Rev. B 89, 035140 (2014)]. As an important result, the density-driven Mott-Hubbard transition has been predicted with a single impurity site, which is a first for this type of embedding methods.
Site-occupation embedding theory (SOET) [B. Senjean et al., Phys. Rev. B 97, 235105 (2018)] is an in-principle exact embedding method combining wave-function theory and density functional theory that gave promising results when applied to the one-dimensional Hubbard model. Despite its overall good performance, SOET faces a computational cost problem as its auxiliary impurity-interacting system remains the size of the full system (which is problematic as the computational cost increases exponentially with system size). In this work, this issue is circumvented by employing the Schmidt decomposition, thus leading to a drastic reduction of the computational cost while retaining the same accuracy. We show that this projected version of SOET (P-SOET) is competitive with other embedding techniques such as density matrix embedding theory (DMET) [G. Knizia and G. K.-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)]. In contrast to the latter, density functional contributions come naturally into play in P-SOET's framework without any additional computational cost or double counting effect. As an important result, the density-driven Mott-Hubbard transition in one dimension (which is displayed by multiple impurity sites in DMET or in dynamical mean-field theory) is well described, for the first time, with a single impurity site.
Site-occupation embedding theory (SOET) is an alternative formulation of density functional theory (DFT) for model Hamiltonians where the fully interacting Hubbard problem is mapped, in principle exactly, onto an impurity-interacting (rather than a noninteracting) one. It provides a rigorous framework for combining wave-function (or Green function)--based methods with DFT. In this work, exact expressions for the per-site energy and double occupation of the uniform Hubbard model are derived in the context of SOET. As readily seen from these derivations, the so-called bath contribution to the per-site correlation energy is, in addition to the latter, the key density functional quantity to model in SOET. Various approximations based on Bethe ansatz and perturbative solutions to the Hubbard and single-impurity Anderson models are constructed and tested on a one-dimensional ring. The self-consistent calculation of the embedded impurity wave function has been performed with the density-matrix renormalization group method. It has been shown that promising results are obtained in specific regimes of correlation and density. Possible further developments have been proposed in order to provide reliable embedding functionals and potentials.
A reformulation of site-occupation embedding theory (SOET) in terms of Green's functions is presented. Referred to as site-occupation Green's-function embedding theory (SOGET), this extension of density-functional theory for model Hamiltonians shares many features with dynamical mean-field theory but is formally exact (in any dimension). In SOGET, the impurity-interacting correlation potential becomes a density-functional self-energy which is frequency dependent and in principle nonlocal. A simple local-density-functional approximation (LDA) combining the Bethe ansatz LDA with the self-energy of the two-level Anderson model is constructed and successfully applied to the one-dimensional Hubbard model. Unlike in previous implementations of SOET, no many-body wave function is needed, thus reducing drastically the computational cost of the method.
Site-occupation embedding theory (SOET) is a density functional theory (DFT)-based method which aims at modelling strongly correlated electrons. It is in principle exact and applicable to model and quantum chemical Hamiltonians. The theory is presented here for the Hubbard Hamiltonian. In contrast to conventional DFT approaches, the site (or orbital) occupations are deduced in SOET from a partially interacting system consisting of one (or more) impurity site(s) and non-interacting bath sites. The correlation energy of the bath is then treated implicitly by means of a site-occupation functional. In this work, we propose a simple impurity-occupation functional approximation based on the two-level (2L) Hubbard model which is referred to as two-level impurity local density approximation (2L-ILDA). Results obtained on a prototypical uniform eight-site Hubbard ring are promising. The extension of the method to larger systems and more sophisticated model Hamiltonians is currently in progress.
The exact formulation of multi-configuration density-functional theory is discussed in this work. As an alternative to range-separated methods, where electron correlation effects are split in the coordinate space, the combination of configuration interaction methods with orbital occupation functionals is explored at the formal level through the separation of correlation effects in the orbital space. When applied to model Hamiltonians, this approach leads to an exact site-occupation embedding theory (SOET). An adiabatic connection expression is derived for the complementary bath functional and a comparison with density matrix embedding theory is made. Illustrative results are given for the simple two-site Hubbard model. SOET is then applied to a quantum chemical Hamiltonian, thus leading to an exact complete active space site-occupation functional theory (CASSOFT) where active electrons are correlated explicitly within the CAS and the remaining contributions to the correlation energy are described with an orbital occupation functional. The computational implementation of SOET and CASSOFT as well as the development of approximate functionals are left for future work.
Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.