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A mathematical analysis of the GW0 method for computing electronic excited energies of molecules

Eric Cancès, David Gontier, Gabriel Stoltz

TL;DR

The paper builds a rigorous mathematical foundation for the GW0 method in finite molecular systems by formulating the GW equations on the imaginary axis via contour deformation and establishing existence and uniqueness of solutions in a perturbative regime. It introduces a comprehensive operator framework, including the particle/hole/time-ordered Green's functions, spectral functions, polarizabilities, and the self-energy, all cast in a robust functional-analytic setting with kernel products and Hilbert transforms. Key contributions include precise definitions and properties of $G$, $\mathcal{A}$, $\chi$, $W$, and $\Sigma$, Johnson-type sum rules, and the analytic continuation results that render the GW0 equations mathematically tractable. The results culminate in a contraction-mapping argument proving well-posedness of the GW0 model near a non-interacting reference, thus providing a rigorous justification for the commonly used G0W0 and GW0 practices in finite systems. Overall, the work bridges rigorous analysis and many-body perturbation theory, offering a solid theoretical basis for GW calculations on molecules and a pathway for future numerical and theoretical developments.

Abstract

This paper analyses the GW method for finite electronic systems. In a first step, we provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. We then discuss the GW equations which construct an approximation of the one-body Green's function, and give a rigorous mathematical formulation of these equations. Finally, we study the well-posedness of the GW0 equations, proving the existence of a unique solution to these equations in a perturbative regime.

A mathematical analysis of the GW0 method for computing electronic excited energies of molecules

TL;DR

The paper builds a rigorous mathematical foundation for the GW0 method in finite molecular systems by formulating the GW equations on the imaginary axis via contour deformation and establishing existence and uniqueness of solutions in a perturbative regime. It introduces a comprehensive operator framework, including the particle/hole/time-ordered Green's functions, spectral functions, polarizabilities, and the self-energy, all cast in a robust functional-analytic setting with kernel products and Hilbert transforms. Key contributions include precise definitions and properties of , , , , and , Johnson-type sum rules, and the analytic continuation results that render the GW0 equations mathematically tractable. The results culminate in a contraction-mapping argument proving well-posedness of the GW0 model near a non-interacting reference, thus providing a rigorous justification for the commonly used G0W0 and GW0 practices in finite systems. Overall, the work bridges rigorous analysis and many-body perturbation theory, offering a solid theoretical basis for GW calculations on molecules and a pathway for future numerical and theoretical developments.

Abstract

This paper analyses the GW method for finite electronic systems. In a first step, we provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. We then discuss the GW equations which construct an approximation of the one-body Green's function, and give a rigorous mathematical formulation of these equations. Finally, we study the well-posedness of the GW0 equations, proving the existence of a unique solution to these equations in a perturbative regime.

Paper Structure

This paper contains 80 sections, 49 theorems, 365 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Setting the stage
  3. Some notation
  4. Hilbert transform of functions and distributions
  5. Hilbert transform in $L^p$ spaces
  6. Hilbert transform in Sobolev spaces
  7. Hilbert transforms of operator-valued distributions
  8. Causal and anti-causal operators
  9. Causal operators
  10. Anti-causal operators
  11. Operators defined by kernel products
  12. Definition of the kernel product
  13. Properties of the kernel product
  14. Laplace transforms of kernel products
  15. Second quantization formalism
  16. ...and 65 more sections

Key Result

Theorem 2

For all $\widehat{f} \in L^p({\mathbb R}_\omega)$ with $1 < p < \infty$, the Hilbert transform is well-defined for almost all $\omega \in {\mathbb R}$. It holds ${\mathfrak H} \in {\mathcal{B}}(L^p({\mathbb R}_\omega))$ with Moreover, the Hilbert transform commutes with the translations and the positive dilations, and anticommutes with the reflexions. Finally, it is a unitary operator on $L^2({\

Figures (6)

  • Figure 1: Illustration of the analytic continuation: from $\omega \mapsto \widehat{G_{\rm p}}(\omega)$ to $\omega \mapsto \widetilde{G_{\rm p}}(\mu + {\mathrm{i}} \omega)$.
  • Figure 2: Illustration of the analytic continuation: from $\omega \mapsto \widehat{G_{\rm h}}(\omega)$ to $\omega \mapsto \widetilde{G_{\rm h}}(\mu + {\mathrm{i}} \omega)$.
  • Figure 3: The cut-off functions $\phi_{\pm}$.
  • Figure 4: The cut-off functions $\phi^1_{\pm}$.
  • Figure 5: The contour $\mathscr{C}$ used in the proof of (iii).
  • ...and 1 more figures

Theorems & Definitions (84)

  • Definition 1: Hilbert transform on $\mathscr{S}({\mathbb R}_\omega)$
  • Theorem 2
  • Lemma 3: Fourier transform in $L^\infty({\mathbb R}_\tau)$
  • Lemma 4
  • Remark 5: Hilbert transform of distributions
  • Definition 6: Hilbert transforms of frequency-dependent operators
  • Lemma 7
  • Definition 8: Principal value of the resolvent of a self-adjoint operator
  • Theorem 9: Titchmarsh's theorem in $L^2$ Titchmarsh_book
  • Theorem 10: Titchmarsh's theorem in $L^\infty({\mathbb R})$
  • ...and 74 more