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Physical Review B

Efficient and stable orbital-searching algorithm for the configuration interaction method and its application to quantum impurity problems

Chungwei Lin, Alexander A. Demkov

Published Dec 11, 2014• 5 citations
Publisher

Abstract

The configuration interaction (CI) method can be regarded as a generalization of the Hartree-Fock theory. The CI method includes multiple determinantal states to approximate the full many-body ground state, as opposed to just one in Hartree-Fock theory. In this work, we introduce an efficient and stable algorithm to obtain the optimal single-particle basis by formulating orbital searching in terms of a multivariable minimization problem. The algorithm is iterative and is based on imaginary-time dynamics. It is numerically efficient as the Hessian (second derivative of the energy with respect to the orbital variables) is not required; it is also stable as a reasonable initial orbital choice is not necessary. The algorithm can deal with Hamiltonians with complex coefficients. We demonstrate the power of the proposed algorithm by applying it to impurity problems, with the most complicated one including multiple correlated orbitals and spin-orbit coupling. Energies and spectral functions are computed to demonstrate the convergence of the CI method.

DOI
10.1103/physrevb.90.235122

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The configuration interaction (CI) method can be regarded as a generalization of the Hartree-Fock theory. The CI method includes multiple determinantal states to approximate the full many-body ground state, as opposed to just one in Hartree-Fock theory. In this work, we introduce an efficient and stable algorithm to obtain the optimal single-particle basis by formulating orbital searching in terms of a multivariable minimization problem. The algorithm is iterative and is based on imaginary-time dynamics. It is numerically efficient as the Hessian (second derivative of the energy with respect to the orbital variables) is not required; it is also stable as a reasonable initial orbital choice is not necessary. The algorithm can deal with Hamiltonians with complex coefficients. We demonstrate the power of the proposed algorithm by applying it to impurity problems, with the most complicated one including multiple correlated orbitals and spin-orbit coupling. Energies and spectral functions are computed to demonstrate the convergence of the CI method.
A direct configuration-interaction (CI) approach for studies of Bose-Einstein condensates has been developed and implemented. The reference state is obtained by solving the Hartree-Fock equation in a nonorthogonal Gaussian basis. The CI problem is solved by diagonalizing the Hamiltonian constructed in a many-body basis generated using the orthogonal canonical orbital basis obtained in a Hartree-Fock calculation. For Hamiltonian matrices, which are too large for full diagonalization, a direct iterative diagonalization scheme is employed for obtaining the lowest-lying eigenstates. The computational methods were tested by applications to Bose-Einstein condensates using various isotropic and anisotropic trap geometries.
Abstract Recent advances in selected configuration interaction methods have made them competitive with the most accurate techniques available and, hence, creating an increasingly powerful tool for solving quantum Hamiltonians. In this work, we build on recent advances from the adaptive sampling configuration interaction (ASCI) algorithm. We show that a useful paradigm for generating efficient selected CI/exact diagonalization algorithms is driven by fast sorting algorithms, much in the same way iterative diagonalization is based on the paradigm of matrix vector multiplication. We present several new algorithms for all parts of performing a selected CI, which includes new ASCI search, dynamic bit masking, fast orbital rotations, fast diagonal matrix elements, and residue arrays. The ASCI search algorithm can be used in several different modes, which includes an integral driven search and a coefficient driven search. The algorithms presented here are fast and scalable, and we find that because they are built on fast sorting algorithms they are more efficient than all other approaches we considered. After introducing these techniques, we present ASCI results applied to a large range of systems and basis sets to demonstrate the types of simulations that can be practically treated at the full-CI level with modern methods and hardware, presenting double- and triple-ζ benchmark data for the G1 data set. The largest of these calculations is Si2H6 which is a simulation of 34 electrons in 152 orbitals. We also present some preliminary results for fast deterministic perturbation theory simulations that use hash functions to maintain high efficiency for treating large basis sets.
Selective configuration interaction methods approximate correlated molecular ground- and excited states by considering only the most relevant Slater determinants in the expansion. While a recently proposed neural-network-assisted approach efficiently identifies such determinants, the procedure typically relies on canonical Hartree-Fock orbitals, which are optimized only at the mean-field level. Here we assess approximate natural orbitals - eigenfunctions of the one-particle density matrix computed from intermediate many-body eigenstates - as an alternative. Across our benchmarks for H2_22​O, NH3_33​, CO, and C3_33​H8_88​ we see a consistent reduction in the required determinants for a given accuracy of the computed correlation energy compared to full configuration interaction calculations. Our results confirm that even approximate natural orbitals constitute a simple yet powerful strategy to enhance the efficiency of neural-network-assisted configuration interaction calculations.
The development of polynomial cost solvers for correlated quantum impurity models, with controllable errors, is a central challenge in quantum many-body physics, where these models find applications ranging from nanoscience to the dynamical mean-field theory (DMFT). Here, we describe how configuration interaction (CI) approximations to exact diagonalization (ED) may be used as solvers in DMFT. CI approximations retain the main advantages of ED, such as the ability to treat general interactions and off-diagonal hybridizations and to obtain real spectral information, but are of polynomial cost. Furthermore, their errors can be controlled by monitoring the convergence of physical quantities as a function of the CI hierarchy. Using benchmark DMFT applications, such as single-site DMFT of the one-dimensional Hubbard model and $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT of the two-dimensional Hubbard model, we show that CI approximations allow us to obtain near-exact ED results for a tiny fraction of the cost. This is true over the entire range of interaction strengths including ``difficult'' regimes, such as in the pseudogap phase of the two-dimensional Hubbard model. We use the ability of CI approximations to treat large numbers of orbitals to demonstrate convergence of the bath representation in the $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT using a 24-bath orbital representation. CI approximations thus form a promising route to extend ED to problems that are currently difficult to study using other solvers such as continuous-time quantum Monte Carlo, including impurity models with large numbers of orbitals and general interactions.
Within the framework of exact diagonalization (ED), we compute the ground state of Anderson impurity problem using the variational approach based on the configuration interaction (CI) expansion. We demonstrate that an accurate ground state can be obtained by iteratively diagonalizing a matrix with the dimension that is less than 10% of the full Hamiltonian. The efficiency of the CI expansion for different problems is analyzed. By way of example, we apply this method to the single-site dynamical mean-field theory using ED as the impurity solver. Specifically, to demonstrate the usefulness of this approach, we solve the attractive Hubbard model in the grand-canonical ensemble, where the sss-wave superconducting solution is explicitly obtained.
A procedure is discussed that searches for the best description of the eigenstates of a Hamiltonian of a finite quantum many-body system in terms of a selected set of physically relevant configurations. The procedure resorts to iterative sequences of diagonalizations in spaces of very reduced size. Each diagonalization provides an energy-based importance measure that governs the selection of the configurations to be included in the states. The procedure is strictly variational and preserves the symmetries of the Hamiltonian throughout the iterative process. We report on some test applications to the Na8{}_{8}8​ metal cluster. A series of calculations is performed on a Hartree-Fock basis for a number of orbitals ranging from 5 to 20. In the case of 5 (10) orbitals, we compare the low-lying spectra generated with our procedure to complete (truncated) configuration interaction calculations. The procedure accurately reproduces the results in the complete space. In the most complex cases of 15 and 20 orbitals (the latter involving a many-particle space of 1011{}^{11}11 states), we construct the ground state and study its structure. The analysis highlights the efficiency of the method in generating the most appropriate model space for the description of individual eigenstates of the Hamiltonian.