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

TL;DR

This paper provides a rigorous mathematical foundation for the density-density response function in TDDFT, demonstrating pole shifting in RPA and justifying the eigenvalue approach used in physics.

Contribution

It establishes existence and uniqueness of solutions to the Dyson equation in TDDFT RPA and mathematically explains the empirical pole shifting phenomenon.

Findings

RPA poles are shifted forward relative to non-interacting poles.

RPA poles are solutions to an eigenvalue problem.

Mathematically justifies the eigenvalue approach in physics.

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.