Coulomb potentials and Taylor expansions in Time-Dependent Density Functional Theory

TL;DR

This paper examines the limitations of using Taylor expansions in Time-Dependent Density Functional Theory, especially with Coulomb potentials, showing that such expansions are generally finite and not suitable for atoms and molecules.

Contribution

It analyzes the impact of Coulomb singularities on the validity of Taylor expansions in TDDFT, revealing fundamental restrictions for their use in atomic and molecular systems.

Findings

Taylor expansions are finite for Coulomb potentials in TDDFT.

High-order Taylor expansions are generally invalid for atoms and molecules.

Singular potentials significantly restrict the applicability of the Runge-Gross theorem.

Abstract

We investigate when Taylor expansions can be used to prove the Runge-Gross Theorem, which is at the foundation of Time-Dependent Density Functional Theory (TDDFT). We start with a general analysis of the conditions for the Runge-Gross argument, especially the time-differentiability of the density. The latter should be questioned in the presence of singular (e.g. Coulomb) potentials. Then, we show that a singular potential in a one-body operator considerably decreases the class of time-dependent external potentials to which the original argument can be applied. A two-body singularity has an even stronger impact and an external potential is essentially incompatible with it. For the Coulomb interaction and all reasonable initial many-body states, the Taylor expansion only exists to a finite order, except for constant external potentials. Therefore, high-order Taylor expansions are not the right tool to study atoms and molecules in TDDFT.