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arXiv:1109.6858 · Physical Review Letters

Effect of Cusps in Time-Dependent Quantum Mechanics

Zeng-hui Yang, Neepa T. Maitra, Kieron Burke

Published Feb 9, 2012• 15 citations
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Abstract

Within nonrelativistic quantum mechanics, spatial cusps in initial wave functions can lead to nonanalytic behavior in time. We suggest a method for calculating the short-time behavior in such situations. For these cases, the density does not match its Taylor expansion in time, but the Runge-Gross proof of time-dependent density functional theory still holds, as it requires only the potential to be time analytic.

DOI
10.1103/physrevlett.108.063003

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Within nonrelativistic quantum mechanics, spatial cusps in initial wave functions can lead to nonanalytic behavior in time. We suggest a method for calculating the short-time behavior in such situations. For these cases, the density does not match its Taylor expansion in time, but the Runge-Gross proof of time-dependent density functional theory still holds, as it requires only the potential to be time analytic.
In the usual treatment of electronic structure, all matter has cusps in the electronic density at nuclei. Cusps can produce nonanalytic behavior in time, even in response to perturbations that are time analytic. We analyze these nonanalyticities in a simple case from many perspectives. We describe a method, the sss expansion, that can be used in several such cases and illustrate it with a variety of examples. These include both the sudden appearance of electric fields and disappearance of nuclei in both one and three dimensions. When successful, the sss expansion yields the dominant short-time behavior, no matter how strong the external electric field, but agrees with linear-response theory in the weak limit. We discuss the relevance of these results to time-dependent density functional theory.
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.
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.
The mapping of time-dependent densities on potentials in systems of identical quantum mechanical particles is examined. This mapping is of significance ever since Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] established its uniqueness, forming the theoretical basis for time-dependent density functional theory. Beyond uniqueness there are two important issues: existence, often called v-representability, and stability. We show that v-representability for localized densities in turn-on situations is not assured and we give a simple example of nonexistence. As for stability, we discuss an inversion procedure and by computing its Lyapunov exponents we demonstrate that the mapping is unstable with respect to fluctuations in the initial state. We argue that such instabilities will plague any inversion procedure.
The Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.