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

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.

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Cited by in corpus (7)

• Non-Adiabatic Approximations in Time-Dependent Density Functional Theory: Progress and Prospects
• A reformulation of time-dependent Kohn-Sham theory in terms of the second time derivative of the density
• A Snapshot of Time-Dependent Density-Functional Theory
• Geometrical Perspective on Spin-Lattice Density-Functional Theory
• Regularity for evolution equations with non-autonomous perturbations in Banach spaces
• A remark on the attainable set of the Schrödinger equation
• Mathematical Elements of Density Functional Theory