Time-dependent V-representability on lattice systems

TL;DR

This paper investigates the limitations of representing time-dependent densities with potentials in lattice systems, revealing non-representability issues due to discretization, and discusses implications for density-functional theory.

Contribution

It identifies and analyzes the breakdown of time-dependent V-representability on lattices, providing examples and discussing the continuum limit where representability is restored.

Findings

Non-representability arises from discretization issues.

Examples provided for two-point and N-point lattices.

Continuum limit restores V-representability.

Abstract

We study the mapping between time-dependent densities and potentials for noninteracting electronic systems on lattices. As discovered recently by Baer [J. Chem. Phys. 128, 044103 (2008)], there exist well-behaved time-dependent density functions on lattices which cannot be associated with any real time-dependent potential. This breakdown of time-dependent V-representability can be tracked down to problems with the continuity equation which arise from discretization of the kinetic-energy operator. Examples are given for lattices with two points and with N points, and implications for practical numerical applications of time-dependent density-functional theory are discussed. In the continuum limit, time-dependent noninteracting V-representability is restored.