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arXiv:1810.12009 · Physical review. B./Physical review. B

Dynamical configuration interaction: Quantum embedding that combines wave functions and Green's functions

Marc Dvorak, Patrick Rinke

Published Mar 25, 2019• 30 citations
arXivHTMLPDF

Abstract

We present the concept, derivation, and implementation of dynamical configuration interaction, a quantum embedding theory that combines Green's function methodology with the many-body wave function. In a strongly correlated active space, we use full configuration interaction (CI) to describe static correlation exactly. We add energy-dependent corrections to the CI Hamiltonian which, in principle, include all remaining correlations derived from the bath space surrounding the active space. Next, we replace the exact Hamiltonian in the bath with one of excitations defined over a correlated ground state. This transformation is naturally suited to the methodology of many-body Green's functions. In this space, we use a modified $GW$/Bethe-Salpeter equation procedure to calculate excitation energies. Combined with an estimate of the ground-state energy in the bath, we can efficiently compute the energy-dependent corrections, which correlate the full set of orbitals, for very low computational cost. We present dimer dissociation curves for ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$ in good agreement with exact results. Additionally, excited states of ${\mathrm{N}}_{2}$ and ${\mathrm{C}}_{2}$ are in excellent agreement with benchmark theory and experiment. By combining the strengths of two disciplines, we achieve a balanced description of static and dynamic correlation in a fully ab initio, systematically improvable framework.

DOI
10.1103/physrevb.99.115134

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•Jan 1, 2022•0 citations
We present the concept, derivation, and implementation of dynamical configuration interaction, a quantum embedding theory that combines Green's function methodology with the many-body wave function. In a strongly correlated active space, we use full configuration interaction (CI) to describe static correlation exactly. We add energy-dependent corrections to the CI Hamiltonian which, in principle, include all remaining correlations derived from the bath space surrounding the active space. Next, we replace the exact Hamiltonian in the bath with one of excitations defined over a correlated ground state. This transformation is naturally suited to the methodology of many-body Green's functions. In this space, we use a modified GWGWGW/Bethe-Salpeter equation procedure to calculate excitation energies. Combined with an estimate of the ground-state energy in the bath, we can efficiently compute the energy-dependent corrections, which correlate the full set of orbitals, for very low computational cost. We present dimer dissociation curves for H2{\mathrm{H}}_{2}H2​ and N2{\mathrm{N}}_{2}N2​ in good agreement with exact results. Additionally, excited states of N2{\mathrm{N}}_{2}N2​ and C2{\mathrm{C}}_{2}C2​ are in excellent agreement with benchmark theory and experiment. By combining the strengths of two disciplines, we achieve a balanced description of static and dynamic correlation in a fully ab initio, systematically improvable framework.
We present a quantum embedding theory called dynamical configuration interaction (DCI) that combines wave function and Green's function theories. DCI captures static correlation in a correlated subspace with configuration interaction and couples to high-energy, dynamic correlation outside the subspace with many-body perturbation theory based on Green's functions. DCI takes the strengths of both theories to balance static and dynamic correlation in a single, fully ab initio embedding concept. The theory adds dynamic correlation around a fixed active space of orbitals with efficient O(N5)\mathcal{O}({N}^{5})O(N5) scaling, while maintaining a multireference treatment of the active space. We show that treating high-energy correlation up to the GWGWGW and Bethe-Salpeter equation level is sufficient even for challenging multireference problems. Our theory treats ground and excited states on equal footing, and we compute the dissociation curve of N2{\mathrm{N}}_{2}N2​, the vertical excitation energies of small molecules, and the ionization spectrum of benzene in excellent agreement with high-level quantum chemistry methods and experiment.
Entangled atoms, exemplified by the two singlet-coupled atoms in a dissociating nitrogen molecule, are key to the unique properties of qubits, quantum sensors, and other quantum materials. Simulating entangled atoms and strong electron correlation is a prerequisite for modeling quantum materials and remains an outstanding challenge for theory. I introduce a correlated wave function approach capable of simulating strong electron correlation in large active spaces. Projected-interacting full configuration interaction plus density functional theory PiFCI+DFT combines near-exact correlated wave functions of multiple partially interacting model systems, each corrected by a formally exact density functional. This approach can access large active spaces and visualize entanglement and strong correlation while maintaining competitive accuracy for molecular properties. I demonstrate PiFCI+DFT for general main-group chemistry, benchmark dissociation curves, and a model for a large entangled-atom system: the spin-singlet ground state of a square lattice of 100 nitrogen atoms. This system’s 300-electron active space is out of reach for existing ab initio multireference methods. PiFCI+DFT accurately recovers the strongly correlated dissociation limit at a modest computational cost, motivating further development for modeling quantum materials.
Quantitative simulation of electronic structure of solids requires treating local and nonlocal electron correlations on an equal footing. We present a new ab initio formulation of Green's function embedding which, unlike dynamical mean-field theory that uses noninteracting bath, derives bath representation with general two-particle interactions in a systematically improvable manner. The resulting interacting-bath dynamical embedding theory (ibDET) utilizes an efficient real-axis coupled-cluster solver to compute the self-energy, approaching the full system limit at much reduced cost. When combined with the GW theory, GW+ibDET achieves good agreement with experimental spectral properties across a range of semiconducting, insulating, and metallic materials. Our approach also enables quantifying the role of nonlocal electron correlation in determining material properties and addressing the long-standing debate on the bandwidth narrowing of metallic sodium.
The self-energy embedding theory (SEET), in which the active space self-energy is embedded in the self-energy obtained from a perturbative method treating the non-local correlation effects, was recently developed in our group. In SEET, the double counting problem does not appear and the accuracy can be improved either by increasing the perturbation order or by enlarging the active space. This method was first calibrated for the 2D Hubbard lattice showing promising results. In this paper, we report an extension of SEET to quantum chemical ab initio Hamiltonians for applications to molecular systems. The self-consistent second-order Green's function method is used to describe the non-local correlations, while the full configuration interaction method is carried out to capture strong correlation within the active space. Using few proof-of-concept examples, we show that SEET yields results of comparable quality to n-electron valence state second-order perturbation theory with the same active space, and furthermore, the full active space can be split into smaller active spaces without further implementation. Moreover, SEET avoids intruder states and does not require any high-order reduced density matrices. These advantages show that SEET is a promising method to describe physical and chemical properties of challenging molecules requiring large active spaces.
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.
Electronic structure of materials from first principles is essential for one-to-one comparisons between theories and experiments. However, the many-body effects resulting from electron correlation and their interplay with other degrees of freedom pose a great challenge to design an ab initio approximate method with good accuracy and efficiency at the same time. In this thesis, we present our recent developments of ab initio many-body theory for solids based on the finite-temperature Green’s function formalism. We begin by reviewing the basic concepts and definitions of realistic electronic Hamiltonians (Ch. 2) as well as Green’s functions at finite temperature (Ch. 3). Overviews of self-consistent many-body perturbation theories and quantum embedding theories are then presented in Chs. 4 and 5 as complementary materials for the following chapters. In Ch. 6, we present algorithms and implementation details for the finite-temperature self-consistent GW (scGW) method in Gaussian-type orbitals for solids. Our implementation is based on the finite-temperature Green’s function formalism in which all equations are solved on the imaginary axis, without resorting to analytical continuation during the self-consistency. No quasiparticle approximation is employed and all matrix elements of self-energies are explicitly evaluated. We show agreement with other, differently formulated finite-temperature scGW implementations. By migrating computationally intensive calculations to GPUs, we obtain scalable results on large supercomputers with nearly optimal performance. Inclusion of relativistic effects in scGW is discussed in Ch. 7. Our formulation of relativistic scGW is based on the exact two-component formalism with one-electron approximation (X2C1e) and non-relativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and their interplay with electron correlation without adjustable parameters. We examine the effect of the X2C1e by comparison to the established four-component formalism and reach excellent agreement. The simplicity of X2C1e enables construction of higher order theories, such as embedding theories, on top of perturbative calculations. In Ch. 8, we use self-energy embedding theory (SEET) to study the spectra of the prototypical undistorted cubic perovskites SrVO3 and SrMnO3. In the strongly correlated metallic SrVO3, we find that the usual attribution of the satellite peaks at -1.8 eV to Hund or Hubbard physics in the t2g orbitals is inconsistent with our calculations. In the strongly correlated insulator SrMnO3 we recover insulating behavior due to a feedback effect between the strongly correlated orbitals and the weakly correlated environment. Our calculation shows a systematic convergence of spectral features as the space of strongly correlated orbitals is enlarged, paving the way to a systematic parameter-free study of correlated perovskites. Lastly, in Ch. 9, we apply the Green’s function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems in strongly correlated solids within the framework of SEET. We describe the implementation details of our Green’s function coupled cluster (GFCC) solver, investigate its performance, and highlight potential advantages and problems on examples of impurities created for antiferromagnetic MnO and paramagnetic SrMnO3. GFCCSD provides satisfactory descriptions for weakly and moderately correlated impurities with sizes that are intractable by existing accurate impurity solvers such as exact diagonalization. However, when correlations become strong, the singles and doubles approximation used in GFCC could lead to instabilities in searching for the particle number present in impurity problems.