On the Mapping of Time-Dependent Densities onto Potentials in Quantum Mechanics

Abstract

The mapping of time-dependent densities on potentials in quantum mechanics is critically examined. The issue is of significance ever since Runge and Gross (Phys. Rev. Lett. 52, 997 (1984)) established the uniqueness of the mapping, forming a theoretical basis for time-dependent density functional theory. We argue that besides existence (so called v-representability) and uniqueness there is an important question of stability and chaos. Studying a 2-level system we find innocent, almost constant densities that cannot be constructed from any potential (non-existence). We further show via a Lyapunov analysis that the mapping of densities on potentials has chaotic regions in this case. In real space the situation is more subtle. V-representability is formally assured but the mapping is often chaotic making the actual construction of the potential almost impossible. The chaotic nature of the mapping, studied for the first time here, has serious consequences regarding the possibility of using TDDFT in real-time settings.

References in corpus (2)

• Optimized Effective Potentials in Finite Basis Sets
• Degeneracy in Density Functional Theory: Topology in v- and n-Space

Cited by in corpus (28)

• Quantum Electrodynamical Density-Functional Theory: Bridging Quantum Optics and Electronic-Structure Theory
• Towards a description of the Kondo effect using time-dependent density functional theory
• Time-dependent V-representability on lattice systems
• Challenging Adiabatic Time-dependent Density Functional Theory with a Hubbard Dimer: The Case of Time-Resolved Long-Range Charge Transfer
• Time-dependent current density functional theory on a lattice
• Existence, Uniqueness, and Construction of the Density-Potential Mapping in Time-Dependent Density-Functional Theory
• Time-dependent density functional theory on a lattice
• Linear optical response of current-carrying molecular junction: A NEGF-TDDFT approach
• Charge-Transfer in Time-Dependent Density Functional Theory: Insights from the Asymmetric Hubbard Dimer
• Non-Adiabatic Approximations in Time-Dependent Density Functional Theory: Progress and Prospects
• Local Control and v-Representability of Correlated Quantum Dynamics
• Linear response time-dependent density functional theory of the Hubbard dimer
• Time-dependent density-functional theory for real-time electronic dynamics on material surfaces
• Some open questions in TDDFT: Clues from Lattice Models and Kadanoff-Baym Dynamics
• Interacting fermions in 1D disordered lattices: Exploring localization and transport properties with lattice density-functional theories
• Nonequilibrium diagrammatic technique for Hubbard Green functions
• Exact time-dependent density functional theory for impurity models
• Adiabatic and local approximations for the Kohn-Sham potential in time-dependent Hubbard chains
• Ultrafast manipulation of electron spins in a double quantum dot device: A real-time view
• Time-dependent bond-current functional theory for lattice Hamiltonians: fundamental theorem and application to electron transport
• Disorder and interactions in systems out of equilibrium: The exact independent-particle picture from density functional theory
• Time-dependent current density functional theory via time-dependent deformation functional theory: A constrained search formulation in the time domain
• Computational complexity of time-dependent density functional theory
• Density Functional Theory of the Hubbard-Holstein Model
• Spectroscopy of the Hubbard dimer: the spectral potential
• Solver for the electronic V-representation problem of time-dependent density functional theory
• On the Action Formalism of Time-dependent Density-functional Theory
• Time-dependent density functional theory for open spin chains