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arXiv:2301.09107 · The Journal of Chemical Physics

A “moment-conserving” reformulation of GW theory

Charles J. C. Scott, Oliver J. Backhouse, George H. Booth

Published Mar 2, 2023• 28 citations
arXivHTMLPDF

Abstract

We show how to construct an effective Hamiltonian whose dimension scales linearly with system size, and whose eigenvalues systematically approximate the excitation energies of GW theory. This is achieved by rigorously expanding the self-energy in order to exactly conserve a desired number of frequency-independent moments of the self-energy dynamics. Recasting GW in this way admits a low-scaling O[N4] approach to build and solve this Hamiltonian, with a proposal to reduce this further to O[N3]. This relies on exposing a novel recursive framework for the density response moments of the random phase approximation, where the efficient calculation of its starting point mirrors the low-scaling approaches to compute RPA correlation energies. The frequency integration of GW, which distinguishes so many different GW variants, can be performed without approximation directly in this moment representation. Furthermore, the solution to the Dyson equation can be performed exactly, avoiding analytic continuation, diagonal approximations, or iterative solutions to the quasiparticle equation, with the full-frequency spectrum obtained from the complete solution of this effective static Hamiltonian. We show how this approach converges rapidly with respect to the order of the conserved self-energy moments and is applied across the GW100 benchmark dataset to obtain accurate GW spectra in comparison to traditional implementations. We also show the ability to systematically converge all-electron full-frequency spectra and high-energy features beyond frontier excitations, as well as avoiding discontinuities in the spectrum, which afflict many other GW approaches.

DOI
10.1063/5.0143291

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The dynamical GW self-energy approximation is a key computational tool to provide the fundamental spectrum of electronic systems. We reformulate this approximation, representing the particle and hole parts of the GW self-energy through a highly compact set of Cayley-transformed moment constraints. The Cayley transformation maps real frequencies to the unit circle, keeping the moments bounded as their order increases, ensuring numerical stability and allowing resolution to be focused on an energy range of interest. We calculate these Cayley-transformed moments via an efficient O[N$^4$] scaling contour integration, and from them, construct a Hermitian upfolded Hamiltonian with a linearly scaling dimensionality with system size. A single-shot diagonalization of this effective Hamiltonian gives an explicit full-frequency G0W0 Green's function with manifestly real poles and non-negative spectral weights. This enables quasiparticle energies, satellite features, and their spectral weights to be obtained across the full G0W0 spectrum. Comparisons with exact G0W0 calculations and convergence across the GW100 test set and the larger Chlorophyll A molecule demonstrate substantially faster and more reliable convergence with moment order than an earlier monomial-moment approach. These Cayley moment representations therefore provide a stable, compact, and systematically improvable route to the complete spectral information of zero-temperature GW, without explicit frequency grids, plasmon-pole models and other common approximations, or analytic continuation.

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By recasting the non-linear frequency-dependent GW quasiparticle equation into a linear eigenvalue problem, we explain the appearance of multiple solutions and unphysical discontinuities in various physical quantities computed within the GW approximation. Considering the GW self-energy as an effective Hamiltonian, it is shown that these issues are key signatures of strong correlation in the (N ± 1)-electron states and can be directly related to the intruder state problem. A simple and efficient regularization procedure inspired by the similarity renormalization group is proposed to avoid such issues and speed up the convergence of partially self-consistent GW calculations.

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Density functional theory (DFT) performs reasonably well for the determination of structural properties of many materials, but fails to predict electronic band gap values accurately. \nSuch a failure of DFT is not unexpected, since there exists no formal justification for interpreting the DFT eigenvalues as addition or removal energies of the many-body system (quasiparticle energies).\nAn alternative approach to the study of exchange-correlation effects in many-particle systems is provided by many-body perturbation theory (MBPT), which defines a rigorous approach to description of excited-state properties based on the Green's function formalism and the concept of quasiparticle electrons. Within MBPT, one can calculate the quasiparticle (QP) energies and the QP amplitudes by approximately solving, within the so-called GW approximation, the set of coupled integro-differential equations proposed by Hedin in 1965.\nThe GW method generally yields significantly better values for QP energies with respect to DFT as it accounts for the dynamic many-body effects in the electron-electron interaction going beyond the mean-field approximation of independent Kohn-Sham particles. The drawback of such an involved approach, however, is its large computational cost, which is mainly due to the evaluation of dielectric matrices, their inversion, and the solution of a non-Hermitian and nonlinear eigenvalue problem. \nThe first part of this PhD research is devoted to the implementation of an efficient and scalable coarse-grained parallel algorithms and the development of state-of-the-art methods to solve the GW equations.\nAll these techniques are then applied to the study of the quasiparticle band structures of SiO2 in the alpha quartz crystalline structure. The effects of the different approximations involved in the theory, the influence of self-consistency, and a systematic analysis of the reliability of the different plasmon-pole models are presented and discussed. \nThe second part of this PhD work is dedicated to the implementation of a methodological approach to the solution of the GW equations based on the so-called projector augmented wave method (PAW) proposed by Blochl in 1994. This new approach permits to remove many of the limitations intrinsic to the use of the pseudopotential technique widely used for ab-initio calculations, allowing one to reach GW results closer to an all-electron implementation while still maintaining computational efficiency, and flexibility. This allows us to achieve a coherent implementation of many different algorithms which enables a detailed comparison of GW calculations not yet performed until now. Results for the quasiparticle band structure and optical spectra of prototype s- and p-compounds are discussed and compared to recent studies reported in the scientific literature (when available).

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Many-body perturbation theory: The GW approximation

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In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.

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Validation of the GreenX library time-frequency component for efficient G W and RPA calculations

Maryam Azizi, Jan Wilhelm, Dorothea Golze +5 more

Electronic structure calculations based on many-body perturbation theory [e.g., GW or the random-phase approximation (RPA)] require function evaluations in the complex time and frequency domain, for example, inhomogeneous Fourier transforms or analytic continuation from the imaginary axis to the real axis. For inhomogeneous Fourier transforms, the time-frequency component of the GreenX library provides time-frequency grids that can be utilized in low-scaling RPA and GW implementations. In addition, the adoption of the compact frequency grids provided by our library also reduces the computational overhead in RPA implementations with conventional scaling. In this paper, we present low-scaling GW and conventional RPA benchmark calculations using the GreenX grids with different codes (FHI-aims, CP2K, and ABINIT) for molecules, two-dimensional materials and solids. Very small integration errors are observed when using 30 time-frequency points for our test cases, namely $<{10}^{\ensuremath{-}8}$ eV/electron for the RPA correlation energies, and $\ensuremath{\le}10$ meV for the $GW$ quasiparticle energies.

Physical review. B./Physical review. B•Jun 3, 2024•16 citations
We show how to construct an effective Hamiltonian whose dimension scales linearly with system size, and whose eigenvalues systematically approximate the excitation energies of GW theory. This is achieved by rigorously expanding the self-energy in order to exactly conserve a desired number of frequency-independent moments of the self-energy dynamics. Recasting GW in this way admits a low-scaling O[N4] approach to build and solve this Hamiltonian, with a proposal to reduce this further to O[N3]. This relies on exposing a novel recursive framework for the density response moments of the random phase approximation, where the efficient calculation of its starting point mirrors the low-scaling approaches to compute RPA correlation energies. The frequency integration of GW, which distinguishes so many different GW variants, can be performed without approximation directly in this moment representation. Furthermore, the solution to the Dyson equation can be performed exactly, avoiding analytic continuation, diagonal approximations, or iterative solutions to the quasiparticle equation, with the full-frequency spectrum obtained from the complete solution of this effective static Hamiltonian. We show how this approach converges rapidly with respect to the order of the conserved self-energy moments and is applied across the GW100 benchmark dataset to obtain accurate GW spectra in comparison to traditional implementations. We also show the ability to systematically converge all-electron full-frequency spectra and high-energy features beyond frontier excitations, as well as avoiding discontinuities in the spectrum, which afflict many other GW approaches.
The dynamical GW self-energy approximation is a key computational tool to provide the fundamental spectrum of electronic systems. We reformulate this approximation, representing the particle and hole parts of the GW self-energy through a highly compact set of Cayley-transformed moment constraints. The Cayley transformation maps real frequencies to the unit circle, keeping the moments bounded as their order increases, ensuring numerical stability and allowing resolution to be focused on an energy range of interest. We calculate these Cayley-transformed moments via an efficient O[N4^44] scaling contour integration, and from them, construct a Hermitian upfolded Hamiltonian with a linearly scaling dimensionality with system size. A single-shot diagonalization of this effective Hamiltonian gives an explicit full-frequency G0W0 Green's function with manifestly real poles and non-negative spectral weights. This enables quasiparticle energies, satellite features, and their spectral weights to be obtained across the full G0W0 spectrum. Comparisons with exact G0W0 calculations and convergence across the GW100 test set and the larger Chlorophyll A molecule demonstrate substantially faster and more reliable convergence with moment order than an earlier monomial-moment approach. These Cayley moment representations therefore provide a stable, compact, and systematically improvable route to the complete spectral information of zero-temperature GW, without explicit frequency grids, plasmon-pole models and other common approximations, or analytic continuation.
By recasting the non-linear frequency-dependent GW quasiparticle equation into a linear eigenvalue problem, we explain the appearance of multiple solutions and unphysical discontinuities in various physical quantities computed within the GW approximation. Considering the GW self-energy as an effective Hamiltonian, it is shown that these issues are key signatures of strong correlation in the (N ± 1)-electron states and can be directly related to the intruder state problem. A simple and efficient regularization procedure inspired by the similarity renormalization group is proposed to avoid such issues and speed up the convergence of partially self-consistent GW calculations.
Density functional theory (DFT) performs reasonably well for the determination of structural properties of many materials, but fails to predict electronic band gap values accurately. \nSuch a failure of DFT is not unexpected, since there exists no formal justification for interpreting the DFT eigenvalues as addition or removal energies of the many-body system (quasiparticle energies).\nAn alternative approach to the study of exchange-correlation effects in many-particle systems is provided by many-body perturbation theory (MBPT), which defines a rigorous approach to description of excited-state properties based on the Green's function formalism and the concept of quasiparticle electrons. Within MBPT, one can calculate the quasiparticle (QP) energies and the QP amplitudes by approximately solving, within the so-called GW approximation, the set of coupled integro-differential equations proposed by Hedin in 1965.\nThe GW method generally yields significantly better values for QP energies with respect to DFT as it accounts for the dynamic many-body effects in the electron-electron interaction going beyond the mean-field approximation of independent Kohn-Sham particles. The drawback of such an involved approach, however, is its large computational cost, which is mainly due to the evaluation of dielectric matrices, their inversion, and the solution of a non-Hermitian and nonlinear eigenvalue problem. \nThe first part of this PhD research is devoted to the implementation of an efficient and scalable coarse-grained parallel algorithms and the development of state-of-the-art methods to solve the GW equations.\nAll these techniques are then applied to the study of the quasiparticle band structures of SiO2 in the alpha quartz crystalline structure. The effects of the different approximations involved in the theory, the influence of self-consistency, and a systematic analysis of the reliability of the different plasmon-pole models are presented and discussed. \nThe second part of this PhD work is dedicated to the implementation of a methodological approach to the solution of the GW equations based on the so-called projector augmented wave method (PAW) proposed by Blochl in 1994. This new approach permits to remove many of the limitations intrinsic to the use of the pseudopotential technique widely used for ab-initio calculations, allowing one to reach GW results closer to an all-electron implementation while still maintaining computational efficiency, and flexibility. This allows us to achieve a coherent implementation of many different algorithms which enables a detailed comparison of GW calculations not yet performed until now. Results for the quasiparticle band structure and optical spectra of prototype s- and p-compounds are discussed and compared to recent studies reported in the scientific literature (when available).
Fully self-consistent GW (sc-GW) – based on the iterative solution of Dyson’s equation – incorporates many-body correlation effects in the electronic ground-state, providing a recipe to consistently describe ground- and excited-states on an equal footing. We have implemented sc-GW in the localized basis all-electron code FHIaims [1]. We show that, for finite systems, the ground-state properties as well as the excitation spectrum of the sc-GW Green’s function are unique and are not affected by different initialization of the sc-GW loop, curing the pathological dependence on the starting point of one-shot GW (G0W0). In addition, self-consistency systematically improves ionization energies and Galitskii-Migdal total energies of closed shell systems compared to G0W0 based on Hartree-Fock and density-functional theory in local and semilocal approximations. Since our implementation is based on interacting Green’s functions (unlike all G0W0 implementations) it is also a first step towards going beyond the GW approach.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
Electronic structure calculations based on many-body perturbation theory [e.g., GW or the random-phase approximation (RPA)] require function evaluations in the complex time and frequency domain, for example, inhomogeneous Fourier transforms or analytic continuation from the imaginary axis to the real axis. For inhomogeneous Fourier transforms, the time-frequency component of the GreenX library provides time-frequency grids that can be utilized in low-scaling RPA and GW implementations. In addition, the adoption of the compact frequency grids provided by our library also reduces the computational overhead in RPA implementations with conventional scaling. In this paper, we present low-scaling GW and conventional RPA benchmark calculations using the GreenX grids with different codes (FHI-aims, CP2K, and ABINIT) for molecules, two-dimensional materials and solids. Very small integration errors are observed when using 30 time-frequency points for our test cases, namely $<{10}^{\ensuremath{-}8}$ eV/electron for the RPA correlation energies, and $\ensuremath{\le}10$ meV for the GWGWGW quasiparticle energies.