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Physical review. B./Physical review. B

Spectral properties from an efficient analytical representation of the G W self-energy within a multipole approximation

Dario A. Leon, Kristian Berland, Cláudia Cardoso

Published May 21, 2025• 3 citations
Publisher

Abstract

We propose an efficient analytical representation of the frequency-dependent $GW$ self-energy $\mathrm{\ensuremath{\Sigma}}$ via a multipole approximation ($\text{MPA-}\mathrm{\ensuremath{\Sigma}}$). The MPA self-energy model is interpolated from a small set of numerical evaluations of $\mathrm{\ensuremath{\Sigma}}$ in the complex frequency plane, similar to the MPA interpolation developed for the screened Coulomb interaction ($\text{MPA-}W$) [D. A. Leon et al., Phys. Rev. B 104, 115157 (2021)]. Crucially, $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ enables a multipole representation for the interacting Green's function $G$ ($\text{MPA-}G$) and, in turn, access to all the spectral properties, including quasiparticle energies (QP) and renormalization factors beyond the linearized QP equation. We validate the $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ and $\text{MPA-}G$ approaches for a diverse set of systems: bulk Si, Na, and Cu, monolayer ${\mathrm{MoS}}_{2}$, the NaCl ion pair, and the ${\mathrm{F}}_{2}$ molecule. We show that, just as for $\text{MPA-}W$, an appropriate choice of frequency sampling in $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ is critical to guarantee computational efficiency and high accuracy. Moreover, the combined $\text{MPA-}W$ and $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range.

DOI
10.1103/physrevb.111.195147

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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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Cambridge University Press eBooks•May 31, 2016•0 citations
We propose an efficient analytical representation of the frequency-dependent GWGWGW self-energy \ensuremathΣ\mathrm{\ensuremath{\Sigma}}\ensuremathΣ via a multipole approximation (MPA-\ensuremathΣ\text{MPA-}\mathrm{\ensuremath{\Sigma}}MPA-\ensuremathΣ). The MPA self-energy model is interpolated from a small set of numerical evaluations of \ensuremathΣ\mathrm{\ensuremath{\Sigma}}\ensuremathΣ in the complex frequency plane, similar to the MPA interpolation developed for the screened Coulomb interaction (MPA-W\text{MPA-}WMPA-W) [D. A. Leon et al., Phys. Rev. B 104, 115157 (2021)]. Crucially, MPA-\ensuremathΣ\text{MPA-}\mathrm{\ensuremath{\Sigma}}MPA-\ensuremathΣ enables a multipole representation for the interacting Green's function GGG (MPA-G\text{MPA-}GMPA-G) and, in turn, access to all the spectral properties, including quasiparticle energies (QP) and renormalization factors beyond the linearized QP equation. We validate the MPA-\ensuremathΣ\text{MPA-}\mathrm{\ensuremath{\Sigma}}MPA-\ensuremathΣ and MPA-G\text{MPA-}GMPA-G approaches for a diverse set of systems: bulk Si, Na, and Cu, monolayer MoS2{\mathrm{MoS}}_{2}MoS2​, the NaCl ion pair, and the F2{\mathrm{F}}_{2}F2​ molecule. We show that, just as for MPA-W\text{MPA-}WMPA-W, an appropriate choice of frequency sampling in MPA-\ensuremathΣ\text{MPA-}\mathrm{\ensuremath{\Sigma}}MPA-\ensuremathΣ is critical to guarantee computational efficiency and high accuracy. Moreover, the combined MPA-W\text{MPA-}WMPA-W and MPA-\ensuremathΣ\text{MPA-}\mathrm{\ensuremath{\Sigma}}MPA-\ensuremathΣ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range.
Spectral properties from an efficient analytical representation of the GWGWGW self-energy within a multipole approximation
We propose an efficient analytical representation of the frequency-dependent GWGWGW self-energy ΣΣΣ via a multipole approximation (MPA-ΣΣΣ). The multipole-Padé model for the self-energy is interpolated from a small set of numerical evaluations of ΣΣΣ in the complex frequency plane, similarly to the previously multipole representation developed for the screened Coulomb interaction (MPA-WWW) [Phys. Rev. B \textbf{104}, 115157 (2021)]. We show that, likewise MPA-WWW, an appropriate choice of frequency sampling in MPA-ΣΣΣ is critical to guarantee computational efficiency and high accuracy. The combined MPA-WWW and MPA-ΣΣΣ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range. Crucially, MPA-ΣΣΣ enables a multipole representation for the interacting Green's function GGG (MPA-GGG), providing a straightforward evaluation of all the spectral properties, and a more general way to define the renormalization factor ZZZ. We validate the MPA-ΣΣΣ and MPA-GGG approaches for diverse systems: bulk Si, Na and Cu, monolayer MoS2_22​, the NaCl ion-pair and the F2_22​ molecule. Moreover, we introduce toy MPA-ΣΣΣ/GGG models to examine the quasiparticle picture in different regimens of weak and strong correlation. With these models, we expose limitations in defining ZZZ from the local derivative of ΣΣΣ.
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.
The GWGWGW approximation is the gold standard for calculating quasiparticle band structures, yet its computational cost frequently necessitates the use of the plasmon-pole approximation (PPA). While PPA is known to be highly accurate for simple metals and weakly correlated semiconductors, its regime of validity in materials with competing energy scales remains poorly quantified. Here, we construct an exact, numerical benchmark of the PPA using the Lehmann representation of the density response on finite one-dimensional Hubbard clusters. By analytically convolving the exact non-interacting Green's function G0G_0G0​ with the exact pole representation of the screened interaction WWW, we compute the GWGWGW self-energy without numerical frequency integration. We demonstrate that in single-scale Mott insulators, the moment-conserving PPA is essentially exact. However, in multi-band semiconductors where interband transitions introduce a low-energy screening channel that competes with high-energy Mott fluctuations, the PPA systematically misjudges the quasiparticle gap by several electron-volts. Furthermore, we show that strong multi-pole screening can drive the exact quasiparticle weight Z→0Z \to 0Z→0, destroying the quasiparticle picture-an effect entirely missed by the PPA, which artificially stabilizes sharp quasiparticles. Finally, we propose a computationally inexpensive diagnostic based on the polydispersity of the loss function's spectral weight, which accurately predicts PPA failure \textit{ab initio}. Our results provide a rigorous framework for assessing the validity of dynamical screening approximations in strongly correlated materials.
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).
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.
Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .