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arXiv:1906.04079 · Journal of Chemical Theory and Computation

Coupled Cluster as an Impurity Solver for Green’s Function Embedding Methods

Avijit Shee, Dominika Zgid

Published Sep 13, 2019• 51 citations
arXivHTMLPDF

Abstract

We investigate the performance of Green's function coupled cluster singles and doubles (CCSD) method as a solver for Green's function embedding methods. To develop an efficient CC solver, we construct the one-particle Green's function from the coupled cluster (CC) wave function based on the non-Hermitian Lanczos algorithm. The major advantage of this method is that its scaling does not depend on the number of frequency points. We have tested the applicability of the CC Green's function solver in the weakly to strongly correlated regimes by employing it for a half-filled one-dimensional (1D) Hubbard model projected onto a single-site impurity problem and a half-filled two-dimensional (2D) Hubbard model projected onto a four-site impurity problem. For the 1D Hubbard model, for all interaction strengths, we observe an excellent agreement with the full configuration interaction (FCI) technique, both for the self-energy and spectral function. For the 2D Hubbard, we have employed an open-shell version of the current implementation and observed some discrepancies from FCI in the strongly correlated regime. Finally, on an example of a small ammonia cluster, we analyze the performance of Green's function CCSD solver within the Self-Energy Embedding Theory (SEET) with Hartree-Fock (HF) and Green's Function second order (GF2) for the treatment of the environment.

DOI
10.1021/acs.jctc.9b00603

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We investigate the performance of Green's function coupled cluster singles and doubles (CCSD) method as a solver for Green's function embedding methods. To develop an efficient CC solver, we construct the one-particle Green's function from the coupled cluster (CC) wave function based on the non-Hermitian Lanczos algorithm. The major advantage of this method is that its scaling does not depend on the number of frequency points. We have tested the applicability of the CC Green's function solver in the weakly to strongly correlated regimes by employing it for a half-filled one-dimensional (1D) Hubbard model projected onto a single-site impurity problem and a half-filled two-dimensional (2D) Hubbard model projected onto a four-site impurity problem. For the 1D Hubbard model, for all interaction strengths, we observe an excellent agreement with the full configuration interaction (FCI) technique, both for the self-energy and spectral function. For the 2D Hubbard, we have employed an open-shell version of the current implementation and observed some discrepancies from FCI in the strongly correlated regime. Finally, on an example of a small ammonia cluster, we analyze the performance of Green's function CCSD solver within the Self-Energy Embedding Theory (SEET) with Hartree-Fock (HF) and Green's Function second order (GF2) for the treatment of the environment.
Within the self-energy embedding theory (SEET) framework, we study the coupled cluster Green's function (GFCC) method in two different contexts: as a method to treat either the system or the environment present in the embedding construction. Our study reveals that when GFCC is used to treat the environment we do not see improvement in total energies in comparison to the coupled cluster method itself. To rationalize this puzzling result, we analyze the performance of GFCC as an impurity solver with a series of transition metal oxides. These studies shed light on the strength and weaknesses of such a solver and demonstrate that such a solver gives very accurate results when the size of the impurity is small. We investigate if it is possible to achieve a systematic accuracy of the embedding solution when we increase the size of the impurity problem. We found that in such a case, the performance of the solver worsens, both in terms of finding the ground state solution of the impurity problem and the self-energies produced. We concluded that increasing the rank of GFCC solver is necessary to be able to enlarge impurity problems and achieve a reliable accuracy. We also have shown that natural orbitals from weakly correlated perturbative methods are better suited than symmetrized atomic orbitals (SAO) when the total energy of the system is the target quantity.
Embedding theories became important approaches used for accurate calculations of both molecules and solids. In these theories, a small chosen subset of orbitals is treated with an accurate method, called an impurity solver, capable of describing higher correlation effects. Ideally, such a chosen fragment should contain multiple orbitals responsible for the chemical and physical behavior of the compound. Handling a large number of chosen orbitals presents a very significant challenge for the current generation of solvers used in the physics and chemistry community. Here, we develop a Green’s function coupled cluster singles doubles and triples (GFCCSDT) solver that can be used for a quantitative description in both molecules and solids. This solver allows us to treat orbital spaces that are inaccessible to other accurate solvers. At the same time, GFCCSDT maintains high accuracy of the resulting self-energy. Moreover, in conjunction with the GFCCSD solver, it allows us to test the systematic convergence of computational studies. Developing the CC family of solvers paves the road to fully systematic Green’s function embedding calculations in solids. In this paper, we focus on the investigation of GFCCSDT self-energies for a strongly correlated problem of SrMnO 3 solid. Subsequently, we apply this solver to solid MnO showing that an approximate variant of GFCCSDT is capable of yielding a high accuracy orbital resolved spectral function.
We describe the use of coupled-cluster theory as an impurity solver in dynamical mean-field theory (DMFT) and its cluster extensions. We present numerical results at the level of coupled-cluster theory with single and double excitations (CCSD) for the density of states and self-energies of cluster impurity problems in the one- and two-dimensional Hubbard models. Comparison to exact diagonalization shows that CCSD produces accurate density of states and self-energies at a variety of values of U/tU/tU/t and filling fractions. However, the low cost allows for the use of many bath sites, which we define by a discretization of the hybridization directly on the real frequency axis. We observe convergence of dynamical quantities using approximately 30 bath sites per impurity site, with our largest 4-site cluster DMFT calculation using 120 bath sites. We suggest that coupled-cluster impurity solvers will be attractive in ab initio formulations of dynamical mean-field theory.
We apply the Green's function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems arising for strongly correlated solids within the self-energy embedding theory (SEET) framework. We describe the details of our GFCC solver implementation, investigate its performance, and highlight potential advantages and problems on examples of impurities created during the self-consistent SEET for antiferromagnetic MnO and paramagnetic SrMnO3\mathrm{Sr}\mathrm{Mn}{\mathrm{O}}_{3}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, our data also shows that 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. These instabilities appear especially severe when the impurity size gets larger and multiple degenerate orbitals with strong correlations are present. We conclude that, to fully check the reliability of GFCCSD results and use them in fully ab initio calculations in the absence of experiments, a verification from a GFCC solver with higher order excitations is necessary.
Abstract We use the cumulant Green’s functions method (CGFM) to study the single-band Hubbard model. The starting point of the method is to diagonalize a cluster (‘seed’) containing N correlated sites and employ the cumulants calculated from the cluster solution to obtain the full Green’s functions for the lattice. All calculations are done directly; no variational or self-consistent process is needed. We benchmark the one-dimensional results for the gap, the double occupancy, and the ground-state energy as functions of the electronic correlation at half-filling and the occupation numbers as functions of the chemical potential obtained from the CGFM against the corresponding results of the thermodynamic Bethe ansatz and the quantum transfer matrix methods. The particle-hole symmetry of the density of states is fulfilled, and the gap, occupation numbers, and ground-state energy tend systematically to the known results as the cluster size increases. We include a straightforward application of the CGFM to simulate the singles occupation of an optical lattice experiment with lithium-6 atoms in an eight-site Fermi-Hubbard chain near half-filling. The method can be applied to any parameter space for one, two, or three-dimensional Hubbard Hamiltonians and extended to other strongly correlated models, like the Anderson Hamiltonian, the t − J, Kondo, and Coqblin-Schrieffer models.
We present a molecular extension of our recently proposed Green’s function embedding method, interacting-bath dynamical embedding theory (ibDET), for computing charged excitation energies at the GW and EOM-CCSD levels. Starting from atom-centered impurities, we construct bath representations that capture the frequency-dependent entanglement between the impurity and its environment and can be systematically improved via the construction of cluster-specific natural orbitals. Utilizing a GW or coupled-cluster Green’s function solver, the self-energy of the full system is assembled from all embedding problems to obtain the interacting Green’s function. We show that ibDET provides accurate spectral properties with much reduced cost for a broad range of systems, including conjugated molecules and nanoclusters. Compared with full-system results, the errors in the predicted ionization potentials and electron affinities are around 0.1 eV or smaller, while each embedding problem includes only a small fraction of the total orbital space. This work provides an efficient and scalable framework for computing spectral properties of molecular systems.