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The density-density response function in time-dependent density functional theory: mathematical foundations and pole shifting

Thiago Carvalho Corso, Mi-Song Dupuy, Gero Friesecke

TL;DR

This work provides a rigorous mathematical foundation for TDDFT in the RPA by proving existence and uniqueness of the Dyson equation solution for the density-density response function, and by establishing that the RPA poles are forward-shifted relative to the non-interacting poles. It introduces a meromorphic extension framework for the Fourier transform of the response, connects pole locations to an eigenvalue problem involving the Hartree operator, and delivers a rigorous Lehmann representation that remains valid in the presence of continuous spectra. The analysis shows that RPA poles can be computed via a reduced eigenproblem and quantifies pole ranks, offering a solid theoretical justification for standard Casida-type approaches used in quantum chemistry to determine excitation frequencies. Overall, the results explain why Kohn–Sham gaps typically underestimate true excitation energies and provide concrete, operator-theoretic criteria for locating and counting RPA poles within the physically relevant spectral window $(-\Omega,\Omega)$.

Abstract

We establish existence and uniqueness of the solution to the Dyson equation for the density-density response function in time-dependent density functional theory (TDDFT) in the random phase approximation (RPA). We show that the poles of the RPA density-density response function are forward-shifted with respect to those of the non-interacting response function, thereby explaining mathematically the well known empirical fact that the non-interacting poles (given by the spectral gaps of the time-independent Kohn-Sham equations) underestimate the true transition frequencies. Moreover we show that the RPA poles are solutions to an eigenvalue problem, justifying the approach commonly used in the physics community to compute these poles.

The density-density response function in time-dependent density functional theory: mathematical foundations and pole shifting

TL;DR

This work provides a rigorous mathematical foundation for TDDFT in the RPA by proving existence and uniqueness of the Dyson equation solution for the density-density response function, and by establishing that the RPA poles are forward-shifted relative to the non-interacting poles. It introduces a meromorphic extension framework for the Fourier transform of the response, connects pole locations to an eigenvalue problem involving the Hartree operator, and delivers a rigorous Lehmann representation that remains valid in the presence of continuous spectra. The analysis shows that RPA poles can be computed via a reduced eigenproblem and quantifies pole ranks, offering a solid theoretical justification for standard Casida-type approaches used in quantum chemistry to determine excitation frequencies. Overall, the results explain why Kohn–Sham gaps typically underestimate true excitation energies and provide concrete, operator-theoretic criteria for locating and counting RPA poles within the physically relevant spectral window .

Abstract

We establish existence and uniqueness of the solution to the Dyson equation for the density-density response function in time-dependent density functional theory (TDDFT) in the random phase approximation (RPA). We show that the poles of the RPA density-density response function are forward-shifted with respect to those of the non-interacting response function, thereby explaining mathematically the well known empirical fact that the non-interacting poles (given by the spectral gaps of the time-independent Kohn-Sham equations) underestimate the true transition frequencies. Moreover we show that the RPA poles are solutions to an eigenvalue problem, justifying the approach commonly used in the physics community to compute these poles.
Paper Structure (26 sections, 23 theorems, 166 equations)

This paper contains 26 sections, 23 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Linear response theory
  3. Time-dependent density functional theory
  4. Main results
  5. Assumptions
  6. Solution to the Dyson equation
  7. Poles of the RPA density-density response function
  8. The ground-state density-density response function
  9. Derivation and Kubo formula
  10. Regularity of the density-density response function
  11. Fourier transform of the density-density response function and its poles
  12. The RPA Dyson equation
  13. Well-posedness of the Dyson equation
  14. Bijection of the RPA-Dyson solution map
  15. Symmetrized density-density response function
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\chi_0 \in C_s\bigr(\mathbb R_+;\mathcal{B}(L^2+L^\infty,L^1\cap L^2)\bigr)$. Then the following assertions are true:

Theorems & Definitions (54)

  • Theorem 1.1: Existence of the solution $\chi^{\mathrm{RPA}}$
  • Theorem 1.2: Poles of $\widehat{\chi^{\textnormal{RPA}}}$
  • Remark 1.3
  • Theorem 1.4: Characterization of the poles of $\widehat{\chi^\mathrm{RPA}}$
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Corollary 2.2: Kubo formula
  • Proposition 2.3
  • proof
  • ...and 44 more