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The Journal of Chemical Physics

Support for the existence of invertible maps between electronic densities and non-analytic 1-body external potentials in non-relativistic time-dependent quantum mechanics

Martín A. Mosquera

Published Oct 4, 2017• 5 citations
Publisher

Abstract

Provided the initial state, the Runge-Gross theorem establishes that the time-dependent (TD) external potential of a system of non-relativistic electrons determines uniquely their TD electronic density, and vice versa (up to a constant in the potential). This theorem requires the TD external potential and density to be Taylor-expandable around the initial time of the propagation. This paper presents an extension without this restriction. Given the initial state of the system and evolution of the density due to some TD scalar potential, we show that a perturbative (not necessarily weak) TD potential that induces a non-zero divergence of the external force-density, inside a small spatial subset and immediately after the initial propagation time, will cause a change in the density within that subset, implying that the TD potential uniquely determines the TD density. In this proof, we assume unitary evolution of wavefunctions and first-order differentiability (which does not imply analyticity) in time of the internal and external force-densities, electronic density, current density, and their spatial derivatives over the small spatial subset and short time interval.

DOI
10.1063/1.4991870

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Provided the initial state, the Runge-Gross theorem establishes that the time-dependent (TD) external potential of a system of non-relativistic electrons determines uniquely their TD electronic density, and vice versa (up to a constant in the potential). This theorem requires the TD external potential and density to be Taylor-expandable around the initial time of the propagation. This paper presents an extension without this restriction. Given the initial state of the system and evolution of the density due to some TD scalar potential, we show that a perturbative (not necessarily weak) TD potential that induces a non-zero divergence of the external force-density, inside a small spatial subset and immediately after the initial propagation time, will cause a change in the density within that subset, implying that the TD potential uniquely determines the TD density. In this proof, we assume unitary evolution of wavefunctions and first-order differentiability (which does not imply analyticity) in time of the internal and external force-densities, electronic density, current density, and their spatial derivatives over the small spatial subset and short time interval.
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 show that a time-dependent particle density n(rt)n(\mathbf{r}t)n(rt) obtained from a given many-particle system can, under mild restrictions on the initial state, always be reproduced by an external potential ${v}^{\ensuremath{'}}(\mathbf{r}t)$ in a many-particle system with different two-particle interactions. Given the initial state of this other many-particle system, the potential ${v}^{\ensuremath{'}}(\mathbf{r}t)$ is unique up to a purely time-dependent function. As a special case we obtain the well-known Runge-Gross theorem.
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.
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.
We reformulate the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed-point question for potentials on a given time interval. We show that the unique fixed point, i.e. the unique potential generating a given density, is reached as the limiting point of an iterative procedure. The one-to-one correspondence between densities and potentials is a straightforward result provided that the response function of the divergence of the internal forces is bounded. The existence, i.e. the v -representability of a density, can be proven as well provided that the operator norms of the response functions of the members of the iterative sequence of potentials have an upper bound. The densities under consideration have second time-derivatives that are required to satisfy a condition slightly weaker than being square-integrable. This approach avoids the usual restrictions of Taylor-expandability in time of the uniqueness theorem by Runge and Gross ( Phys. Rev. Lett. , 52 (1984) 997 ) and of the existence theorem by van Leeuwen ( Phys. Rev. Lett. , 82 (1999) 3863 ). Owing to its generality, the proof not only answers basic questions in density-functional theory but also has potential implications in other fields of physics.
This work studies in detail the possibility of defining a one-to-one mapping from charge densities as obtained by the time-dependent Schrödinger equation to external potentials. Such a mapping is provided by the Runge-Gross theorem and lies at the very core of time-dependent density functional theory. After introducing the necessary mathematical concepts, the usual mapping "there" - from potentials to wave functions as solutions to the Schrödinger equation - is revisited paying special attention to Sobolev regularity. This is scrutinised further when the question of functional differentiability of the solution with respect to the potential arises, a concept related to linear response theory. Finally, after a brief introduction to general density functional theory, the mapping "back again" - from densities to potentials thereby inverting the Schrödinger equation for a fixed initial state - is defined. Apart from utilising the original Runge-Gross proof this is achieved through a fixed-point procedure. Both approaches give rise to mathematical issues, previously unresolved, which however could be dealt with to some extent within the framework at hand.