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The European Physical Journal B

Numerical construction of the density-potential mapping

S. E. B. Nielsen, Michael Ruggenthaler, Robert van Leeuwen

Published Oct 1, 2018• 14 citations
Publisher

Abstract

We demonstrate how a recently developed method Nielsen et al. [Nielsen et al., EPL 101, 33001 (2013)] allows for a comprehensive investigation of time-dependent density functionals in general, and of the exact time-dependent exchange-correlation potential in particular, by presenting the first exact results for two- and three-dimensional multi-electron systems. This method is an explicit realization of the Runge–Gross correspondence, which maps time-dependent densities to their respective potentials, and allows for the exact construction of desired density functionals. We present in detail the numerical requirements that makes this method efficient, stable and precise even for large and rapid density changes, irrespective of the initial state and two-body interaction. This includes among others the proper treatment of low density regions, a subtle interplay between numerical time-propagation and zero boundary conditions, the choice of time-stepping strategy, and an error damping mechanism based on both the density and current density residuals. These considerations are also relevant for computing time-independent density-functionals and lead to a more precise implementation of quantum mechanics in general, which is mainly relevant for cases in which there is notable contact with a boundary or when the low density regions matter.

DOI
10.1140/epjb/e2018-90276-4

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We demonstrate how a recently developed method Nielsen et al. [Nielsen et al., EPL 101, 33001 (2013)] allows for a comprehensive investigation of time-dependent density functionals in general, and of the exact time-dependent exchange-correlation potential in particular, by presenting the first exact results for two- and three-dimensional multi-electron systems. This method is an explicit realization of the Runge–Gross correspondence, which maps time-dependent densities to their respective potentials, and allows for the exact construction of desired density functionals. We present in detail the numerical requirements that makes this method efficient, stable and precise even for large and rapid density changes, irrespective of the initial state and two-body interaction. This includes among others the proper treatment of low density regions, a subtle interplay between numerical time-propagation and zero boundary conditions, the choice of time-stepping strategy, and an error damping mechanism based on both the density and current density residuals. These considerations are also relevant for computing time-independent density-functionals and lead to a more precise implementation of quantum mechanics in general, which is mainly relevant for cases in which there is notable contact with a boundary or when the low density regions matter.
In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schrödinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain Sturm-Liouville problem, which we discuss for different situations. Based on these considerations we then present a discussion of the famous Runge-Gross theorem which provides a density-potential mapping for time-analytic potentials. Further we give conditions such that the general fixed-point approach is well-defined and converges under certain assumptions. Then the application of such a fixed-point procedure to lattice Hamiltonians is discussed and the numerical realization of the density-potential mapping is shown. We conclude by presenting an extension of the density-potential mapping to include vector-potentials and photons.
We present a local control scheme to construct the external potential v that, for a given initial state, produces a prescribed time-dependent density in an interacting quantum many-body system. This numerical method is efficient and stable even for large and rapid density variations irrespective of the initial state and the interactions. It can at the same time be used to answer fundamental v -representability questions in density functional theory. In particular, in the absence of interactions, it allows us to construct the exact time-dependent Kohn-Sham potential for arbitrary initial states. We illustrate the method in a correlated one-dimensional two-electron system with different interactions, initial states and densities. For a Kohn-Sham system with a correlated initial state we demonstrate the interplay between memory and initial-state dependence as well as the failure of any adiabatic approximation.
In a recent paper [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed-point problem. This idea was used to generalize the existence and uniqueness theorems underlying time-dependent density-functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case because it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that, in the most physically relevant cases, the fixed-point procedure converges. This is further demonstrated with an example.
This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is v-representable. Then the lattice TDDFT formulation is extended to cover system of interacting lattice electrons strongly coupled to cavity photons.We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v representability theorem to include the ground state of a general lattice electron-photon Hamiltonian.
Under reasonable assumptions the time-dependent particle density $n(\stackrel{P\vec}{r},t)$ and the current density $\stackrel{P\vec}{j}(\stackrel{P\vec}{r},t)$ of a many-particle system that evolves under the action of external scalar and vector potentials $V(\stackrel{P\vec}{r},t)$ and $\stackrel{P\vec}{A}(\stackrel{P\vec}{r},t)$ and is initially in the quantum state $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$ can be reproduced in another many-particle system with a different two-particle interaction, subjected to external potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and starting from an initial state $\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{'}}(0)⟩$, which yields the same density and current as $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$. Here we show that given the initial state of this other many-particle system, the potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$, if they exist, are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time-dependent current density functional theory to transport problems.
In this work we focus on a recently introduced method [1] to construct the external potential vvv that, for a given initial state, produces a prescribed time-dependent density in an interacting quantum many-body system. We show how this method can also be used to perform flexible and efficient quantum control. The simple interpretation of the density (the amount of electrons per volume) allows us to use our physical intuition to consider interesting control problems and to easily restrict the search space in optimization problems. The method's origin in time-dependent density-functional theory makes studies of large systems possible. We further discuss the generalization of the method to higher dimensions and its numerical implementation in great detail. We also present several examples to illustrate the flexibility, and to confirm that the scheme is efficient and stable even for large and rapid density variations irrespective of the initial state and interactions.