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A mathematical analysis of the adiabatic Dyson equation from time-dependent density functional theory

Thiago Carvalho Corso

TL;DR

This work provides a rigorous functional-analytic treatment of the adiabatic Dyson equation in LR-TDDFT, unifying discrete and continuum settings and enabling a representation of the interacting DDRF via an operator Casida matrix. The authors derive a precise Fourier-domain formula $\widehat{\chi_F}(z) = 2B H_#^{1/2} (z^2-\mathcal{C})^{-1} H_#^{1/2} B^*$, with the Casida operator $\mathcal{C} = H_#^2 + 2 H_#^{1/2} B^* F B H_#^{1/2}$, and provide a time-domain expression involving $\mathrm{sinc}(t\sqrt{\mathcal{C}})$. They prove well-posedness of the Dyson equation, analyze stability through the auxiliary operator $\mathcal{M}$, and establish that, for common adiabatic approximations such as RPA and ALDA, the maximal meromorphic domain of $\widehat{\chi_F}$ coincides with that of $\widehat{\chi_H}$, implying these approximations do not shift the ionization threshold. The results yield practical criteria for admissible adiabatic kernels, clarify the spectral structure of the response, and connect to absorption spectra via the polarizability operator, enhancing the mathematical foundation of LR-TDDFT applications.

Abstract

In this article, we analyze the Dyson equation for the density-density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterize the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial density-density response function and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation (RPA) and the adiabatic local density approximation (ALDA). In particular, these results show that neither of these approximations can shift the ionization threshold of the Kohn-Sham system.

A mathematical analysis of the adiabatic Dyson equation from time-dependent density functional theory

TL;DR

This work provides a rigorous functional-analytic treatment of the adiabatic Dyson equation in LR-TDDFT, unifying discrete and continuum settings and enabling a representation of the interacting DDRF via an operator Casida matrix. The authors derive a precise Fourier-domain formula , with the Casida operator , and provide a time-domain expression involving . They prove well-posedness of the Dyson equation, analyze stability through the auxiliary operator , and establish that, for common adiabatic approximations such as RPA and ALDA, the maximal meromorphic domain of coincides with that of , implying these approximations do not shift the ionization threshold. The results yield practical criteria for admissible adiabatic kernels, clarify the spectral structure of the response, and connect to absorption spectra via the polarizability operator, enhancing the mathematical foundation of LR-TDDFT applications.

Abstract

In this article, we analyze the Dyson equation for the density-density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterize the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial density-density response function and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation (RPA) and the adiabatic local density approximation (ALDA). In particular, these results show that neither of these approximations can shift the ionization threshold of the Kohn-Sham system.
Paper Structure (17 sections, 22 theorems, 155 equations)

This paper contains 17 sections, 22 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The density-density response function
  4. Well-posedness of the Dyson equation
  5. The Dyson density-density response function
  6. Applications
  7. Mathematical background
  8. Sobolev scale of spaces
  9. Quadratic forms
  10. Reducing subspaces and block decomposition
  11. Proofs
  12. Proof of Theorem \ref{['thm:chiH']}
  13. Proof of Theorem \ref{['thm:Casidarep']}
  14. Proof of Corollaries \ref{['cor:chiFtime']} to \ref{['cor:maximal']}
  15. Proof of Theorem \ref{['thm:maximaldomain']}
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $H$ be a Hamiltonian satisfying Assumption assump:main. Then the maximal meromorphic (see Definition def:meromorphic) extension of the Fourier transform of $\chi_H$ is given by where In particular, the set of poles of $\widehat{\chi_H}$ is given by $\sigma^{\mathrm{disc}}_1(H_\#)\cup \sigma_1^{\mathrm{disc}}(-H_\#)$. Moreover, these poles are all simple, and their rank is given by

Theorems & Definitions (53)

  • Remark : Response theory terminology
  • Theorem 2.1: Maximal meromorphic extension
  • Remark : Poles with infinite rank
  • Remark : Dark excitations
  • Theorem 2.2: Well-posedness of the Dyson equation
  • Theorem 2.3: Solution in the frequency domain
  • Corollary 2.4: Solution in the time domain
  • Remark
  • Corollary 2.5: Stability condition
  • Remark : Quantitative stability
  • ...and 43 more